Exercise 5.3[br][img]data:image/png;base64,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[/img][br][br]1. (a) Express [math] 2\sin A \cos B [/math] into the sum or difference of Sine or Cosine.[br]Solution:[br][math] 2\sin A \cos B = \sin (A+B) + \sin (A-B) [/math][br][br]1. (b) Express [math] 2\cos A \cos B [/math] into the sum or difference of Cosine.[br]Solution:[br][math]2\cos A \cos B = \cos (A+B) + \cos (A - B) [/math][br][br]1. (c) Reduce [math]2\sin x \sin y [/math] into the sum or difference of Cosine.[br]Solution:[br][math]2\sin x \sin y = 2\cos (x-y) - 2\cos (x+y) [/math][br][br]2. (a) Express [math] \sin C + \sin D [/math] into the product form of Sine and Cosine.[br]Solution:[br][math] \sin C + \sin D = 2\sin \left( \frac{C+D}{2}\right) \cos \left( \frac{C-D}{2}\right) [/math][br][br]2. (b) Express [math] \cos \alpha - \cos \beta [/math] into the product form of sine. [br]Solution:[br][math] \cos \alpha - \cos \beta = 2\sin\left( \frac{\alpha + \beta }{2}\right)\sin\left( \frac{\beta - \alpha }{2}\right) [/math][br][br]3.(a) If [math] \sin (A+B) = \frac{1}{4} [/math] and [math] \sin (A-B) = \frac{3}{4} [/math], find the value of [math] 2\sin A. \cos B [/math].[br]Solution:[br][math] \begin{align}[br] 2\sin A . \cos B \ & = \sin (A+B) - \sin(A-B) \\[br] \ & = \frac{1}{4} + \frac{3}{4} \\[br] \ & = \frac{1+3}{4}\\[br] \ & = \frac{4}{4}\\[br] \ & = 1 \\[br]\end{align} [/math][br][br]3.(b) If [math] \cos (A-B) = \frac{2}{3} [/math] and [math] \cos (A+B) = \frac{1}{3} [/math], find the value of [math] 2\cos A. \cos B [/math].[br]Solution:[br][math] \begin{align}[br] 2\cos A . \cos B \ & = \cos (A-B) + \sin(A+B) \\[br] \ & = \frac{2}{3} + \frac{1}{3} \\[br] \ & = \frac{2+1}{3}\\[br] \ & = \frac{3}{3}\\[br] \ & = 1 \\[br]\end{align} [/math][br][br]4. (a) Find the value of: [math] \cos 70^{\circ} - \cos40^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \ \ \cos 70^{\circ} - \cos40^{\circ}\\[br] \ & = 2\sin \frac{70^{\circ}+40^{\circ}}{2} \sin \frac{40^{\circ}-70^{\circ}}{2}\\[br] \ & =2 \sin \frac{110^{\circ}}{2} \sin \frac{-30^{\circ}}{2}\\[br] \ & = 2 \sin 55^{\circ}\sin (-15^{\circ})\\[br] \ & = -2\sin 15^{\circ} \sin 55^{\circ}\\[br]\end{align} [/math][br]4. (b) Find the value of: [math] \cos 15^{\circ} - \cos75^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \ \ \cos 15^{\circ} - \cos75^{\circ}\\[br] \ & = 2\sin \frac{15^{\circ}+75^{\circ}}{2} \sin \frac{75^{\circ}-15^{\circ}}{2}\\[br] \ & =2 \sin \frac{90^{\circ}}{2} \sin \frac{60^{\circ}}{2}\\[br] \ & = 2 \sin 45^{\circ}\sin (30^{\circ})\\[br] \ & = 2\times \frac{1}{\sqrt{2}}\times \frac{1}{2}\\[br] \ & = \frac{1}{\sqrt{2}}\\[br]\end{align} [/math][br]4. (c) Find the value of: [math] \cos 105^{\circ} + \cos15^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \ \ \cos 105^{\circ} + \cos15^{\circ}\\[br] \ & = 2\cos \frac{105^{\circ}+15^{\circ}}{2} \cos \frac{105^{\circ}-15^{\circ}}{2}\\[br] \ & =2 \cos \frac{120^{\circ}}{2} \cos \frac{90^{\circ}}{2}\\[br] \ & = 2 \cos 60^{\circ}\cos 45^{\circ} \\ \ & = 2\times \frac{1}{2}\times \frac{1}{\sqrt{2}}\\[br] \ & = \frac{1}{\sqrt{2}}\\[br]\end{align} [/math][br]4. (d) Find the value of: [math] \sin 75^\circ \sin 15^{\circ} [/math][br]Solution: [br][math] \begin{align}[br] \ & \sin 75^\circ \sin 15^{\circ} \\[br] \ & = \frac{1}{2} [2\sin 75^\circ \sin 15^{\circ}]\\[br] \ & = \frac{1}{2} [\cos (75^\circ - 15^\circ) - \cos (75^\circ+15^\circ)]\\[br] \ & = \frac{1}{2}[ \cos 60^\circ - \cos 90^\circ ]\\[br] \ & = \frac{1}{2} \left[ \frac{1}{2} - 0 \right]\\[br] \ & = \frac{1}{2} \left[ \frac{1}{2} \right]\\[br] \ & = \frac{1}{2} \times \frac{1}{2}\\[br] \ & = \frac{1}{4}[br] \end{align} [/math][br]4. (e) Find the value of: [math]4 \sin 105^{\circ} \sin15^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & 4 \sin 105^{\circ} \sin15^{\circ}\\[br] \ & = 2 \left[ 2 \sin 105^{\circ} \sin15^{\circ}\right]\\[br] \ & = 2 \left[ \cos (105^{\circ}-15^{\circ} - \cos (105^{\circ}+ 15^{\circ} ) \right]\\[br] \ & = 2 \left( \cos 90^{\circ} - \cos 120^{\circ} \right)\\[br] \ & = 2 \left( 0 + \frac{1}{2} \right)\\[br] \ & = 2\times \frac{1}{2}\\[br] \ & = 1\\[br] \ & = \text{RHS}\\[br]\end{align} [/math][br]4. (f) Find the value of: [math] \cos 15^{\circ} \cos105^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \cos 15^{\circ} \cos105^{\circ}\\[br] \ & = \frac{1}{2} [2\cos 15^{\circ} \cos105^{\circ} ]\\[br] \ & = \frac{1}{2} [\cos (15^{\circ}+105^{\circ}) + \cos (15^{\circ}-105^{\circ})]\\[br] \ & = \frac{1}{2}[ \cos 120^{\circ} + \cos (-90^{\circ}) ]\\[br] \ & = \frac{1}{2}\left[ \frac{-1}{2}+ \cos 90^{\circ}\right]\\[br] \ & = \frac{1}{2}\left[ \frac{-1}{2} + 0 \right]\\[br] \ & = \frac{1}{2}\left[ \frac{-1}{2} \right]\\[br] \ & = \frac{-1}{4}\\[br]\end{align} [/math][br]5. (a) Express the following product into sum or difference form:[br] [math] \sin 61^{\circ} \sin43^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \sin 61^{\circ} \sin43^{\circ}\\[br] \ & = \frac{1}{2}[2\sin 61^{\circ} \sin43^{\circ}]\\[br] \ & = \frac{1}{2} [ \cos (61^{\circ}-43^{\circ}) - \cos (61^{\circ} + 43^{\circ} )]\\[br] \ & = \frac{1}{2}[\cos 18^{\circ} - \cos 104^{\circ} ]\\[br]\end{align} [/math][br]5. (b) Express the following product into sum or difference form:[br] [math] \sin 36^{\circ} \sin24^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \sin 36^{\circ} \sin24^{\circ}\\[br] \ & = \frac{1}{2} [ 2\sin 36^{\circ} \sin24^{\circ}]\\[br] \ & = \frac{1}{2} [\cos (36^{\circ}- 24^{\circ}) - \cos ( 36^{\circ}+24^{\circ})]\\[br] \ & = \frac{1}{2}[ \cos 12^{\circ}-\cos 60^{\circ}]\\[br] \ & = \frac{1}{2}\left[ \cos 12^{\circ}- \frac{1}{2}\right]\\[br]\end{align} [/math][br]5. (c) Express the following product into sum or difference form:[br] [math] \cos 140^{\circ} \cos40^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & \cos 140^{\circ} \cos40^{\circ} \\[br] \ & = \frac{1}{2} \left[ 2\cos 140^{\circ} \cos40^{\circ} \right]\\[br] \ & = \frac{1}{2} \left[ \cos (140^{\circ}+40^{\circ})+\cos (140^{\circ}-40^{\circ}) \right]\\[br] \ & = \frac{1}{2} \left[ \cos 180^{\circ} + \cos 100^{\circ} \right]\\[br] \ & = \frac{1}{2} \left[ -1 + \cos 100^{\circ} \right]\\[br] \ & = \frac{1}{2} \left[ \cos 100^{\circ} -1 \right][br]\end{align} [/math][br]5. (d) Express the following product into sum or difference form:[br] [math] 2\sin 48^{\circ} \cos32^{\circ} [/math][br]Solution:[br][math] \begin{align}[br] \ & 2\sin 48^{\circ} \cos32^{\circ}\\[br] \ & = \sin (48^{\circ}+32^{\circ}) + \cos(48^{\circ}-32^{\circ})\\[br] \ & = \sin 80^{\circ} + \cos 16^{\circ} \\[br]\end{align} [/math][br]5. (e) Express the following product into sum or difference form:[br] [math] 2\sin 5\theta \cos 2\theta [/math][br]Solution:[br][math] \begin{align}[br] \ & 2\sin 5\theta \cos 2\theta\\[br] \ & = \sin (5\theta +2\theta) + \sin (5\theta -2\theta)\\[br] \ & = \sin 7\theta + \sin 3\theta \\[br]\end{align} [/math][br]5. (f) Express the following product into sum or difference form.:[br] [math] 2\sin 9\theta . \cos 7\theta [/math][br]Solution:[br][math] \begin{align}[br] \ & \sin 9\theta \cos 7\theta \\[br] \ & = \frac{1}{2}[2\sin 9\theta \cos 7\theta] \\[br] \ & = \frac{1}{2}[\sin (9\theta + 7 \theta ) + \sin (9\theta -7\theta)] \\[br] \ & = \frac{1}{2}[\sin 16\theta + \sin 2\theta] \\[br]\end{align} [/math][br]5. (g) Express the following product into sum or difference form:[br] [math] 2\cos 11\theta . \cos 3\theta [/math][br]Solution:[br][math] \begin{align}[br] \ & 2\cos 11\theta . \cos 3\theta \\[br] \ & = \cos (11\theta + 3\theta) + \cos (11\theta -3\theta) \\[br] \ & = \cos 14\theta + \cos 8 \theta \\[br]\end{align} [/math][br]5. (h) Express the following product into sum or difference form:[br] [math] 2\sin 7\theta . \sin 3\theta [/math][br]Solution:[br][math] \begin{align}[br] \ & 2\sin 7\theta . \sin 3\theta \\[br] \ & = \cos (7\theta -3\theta) - \cos (7\theta +3\theta) \\[br] \ & = \cos 4\theta -\cos 10\theta \\[br]\end{align} [/math][br][br]6. (a) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ & \cos 65^{\circ} + \cos 25^{\circ}\\[br]\ & = 2\cos \frac{ 65^{\circ}+ 25^{\circ}}{2}\cos \frac{ 65^{\circ}- 25^{\circ}}{2} \\[br]\ & = 2\cos \frac{ 90^{\circ}}{2} \cos \frac{40^{\circ}}{2} \\[br]\ & = 2\cos 45^{\circ} \cos 20^{\circ} \\[br]\ & = 2\times \frac{1}{\sqrt{2}} \cos 20^{\circ} \\[br]\ & = (\sqrt{2})^2 \times \frac{1}{\sqrt{2}} \cos 20^{\circ} \\[br]\ & = \sqrt{2} \cos 20^{\circ} \\[br]\end{align} [/math][br][br]6. (b) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ & \sin 46^{\circ} - \sin 20^{\circ}\\[br]\ & = 2\cos \frac{46^{\circ}+20^{\circ}}{2}\sin \frac{46^{\circ}-20^{\circ}}{2}\\[br]\ & = 2 \cos \frac{66^{\circ}} {2} \sin \frac{ 26^{\circ}}{2}\\[br]\ & = 2 \cos 33^{\circ} \sin 13^{\circ}\\[br]\end{align} [/math][br][br]6. (c) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ & \cos 70^{\circ} + \cos 30^{\circ}\\[br]\ & = 2\cos \frac{ 70^{\circ}+ 30^{\circ}}{2}\cos \frac{ 70^{\circ}- 30^{\circ}}{2} \\[br]\ & = 2\cos \frac{ 100^{\circ}}{2} \cos \frac{40^{\circ}}{2} \\[br]\ & = 2\cos 50^{\circ} \cos 20^{\circ} \\[br]\end{align} [/math][br][br]6. (d) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ & \sin 7\theta- \sin 3\theta\\[br]\ & = 2\cos \frac{7\theta+3\theta}{2}\sin \frac{7\theta-3\theta}{2}\\[br]\ & = 2 \cos \frac{10\theta} {2} \sin \frac{4\theta}{2}\\ [br]\ & = 2 \cos 5\theta \sin 2\theta [br]\end{align} [/math][br][br]6. (e) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ &\sin 11\theta + \sin 3\theta\\[br]\ & = 2\sin \frac{11\theta +3\theta}{2} \cos \frac{11\theta - 3\theta }{2} \\[br]\ & = 2\sin \frac{14\theta}{2} \cos \frac{8\theta}{2}\\[br]\ & = 2\sin 7\theta \cos 4\theta\\[br]\end{align} [/math][br][br]6. (f) Express the following sum or difference into product form.[br]Solution:[br][math] \begin{align}[br]\ & \cos 15\alpha + \cos 5\alpha \\[br]\ & = \cos \frac{ 15\alpha +5\alpha}{2} \cos \frac{15\alpha -5\alpha}{2}\\[br]\ & = \cos \frac{20\alpha}{2} \cos \frac{10\alpha}{2}\\[br]\ & = \cos 10\alpha \cos 5\alpha\\[br]\end{align} [/math][br][br]7. (a) Prove that: [math] \sin 5A +\sin 7A = \sin 6A \cos A [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS}\ &=\sin 5A +\sin 7A \\[br]\ & = \sin \frac{5A+7A}{2}\cos \frac{5A -7 A}{2} \\[br]\ & = \sin \frac{12A }{2} \cos \frac{-2A}{2}\\[br]\ & = \sin 6A \cos (-A)\\[br]\ & = \sin 6A \cos A \\[br]\ & = \text{ RHS}[br]\end{align} [/math][br][br]7. (b) [math] \cos A + \cos 7A = \cos 4A \cos 3A [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & =\cos A + \cos 7A \\[br]\ & = \cos \frac{A+7A}{2} \cos \frac{A-7A}{2}\\[br]\ & = \cos \frac{8A}{2} \cos \frac{-6A}{2}\\[br]\ & = \cos 4A \cos (-3A) \\[br]\ & = \cos 4A \cos 3A \\[br]\ & = \text{ RHS}[br]\end{align} [/math][br][br]7(c) [math] \frac{\cos B -\cos A}{\cos A + \cos B} = \tan \frac{A+B}{2} \tan \frac{A-B}{2} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\cos B -\cos A}{\cos A + \cos B}\\[br]\ & = \frac{ 2\sin \frac{A+B}{2} \sin \frac{A-B}{2} }{2\cos \frac{A+B}{2} \cos \frac{A-B}{2}} \\[br]\ & =\frac{\sin \frac{A+B}{2}}{\cos \frac{A+B}{2}}\times \frac{\sin \frac{A-B}{2}}{\cos \frac{A-B}{2}}\\[br]\ & = \tan \frac{A+B}{2} \tan \frac{A-B}{2}\\[br]\ & =\text{RHS}[br]\end{align} [/math][br][br]7. (d) [math] \frac{\sin A + \sin B}{\sin A - \sin B} = \tan \frac{A+B}{2} \cot \frac{A-B}{2} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS}\ & =\frac{\sin A + \sin B}{\sin A - \sin B}\\[br]\ & = \frac{2\sin \frac{A+B}{2} \cos \frac{A-B}{2}}{2\cos \frac{A+B}{2}\sin \frac{A-B}{2}}\\[br]\ &=\frac{ \sin \frac{A+B}{2}}{\cos \frac{A+B}{2}} \times \frac {\cos \frac{A-B}{2}}{\sin \frac{A-B}{2}}\\[br]\ & = \tan \frac{A+B}{2} \cot \frac{A-B}{2}\\[br]\end{align} [/math][br][br]7. (e) [math] \sin 4\theta \cos 2\theta + \cos 3\theta \sin \theta = \sin 5\theta \cos \theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS}\ & =\sin 4\theta \cos 2\theta + \cos 3\theta \sin \theta \\[br]\ & =\frac{1}{2} [2\sin 4\theta \cos 2\theta + 2\cos 3\theta \sin \theta ]\\[br]\ & = \frac{1}{2} \left[ \sin (4\theta + 2\theta) + \sin (4\theta - 2\theta)+ \sin (3\theta + \theta) - \sin (3\theta - \theta) \right] \\[br]\ & = \frac{1}{2} [ \sin 6\theta + \sin 2\theta + \sin 4\theta -\sin 2\theta ]\\[br]\ & = \frac{1}{2} [ \sin 6\theta + \sin 4\theta ]\\[br]\ & = \frac{1}{2} \left[ 2\sin \frac{6\theta +4\theta}{2} \cos \frac{6\theta - 4\theta}{2} \right] \\[br]\ & =\sin \frac{10\theta}{2} \cos \frac{2\theta}{2} \\[br]\ & = \sin 5\theta \cos \theta \\[br]\ & = \text{ RHS}\\[br]\end{align} [/math][br][br]7. (f) [math] \sin \frac{11\alpha}{4} \sin \frac{\alpha}{4} + \sin \frac{7\alpha}{4} \sin \frac{3\alpha}{4} = \sin 2\alpha \sin \alpha [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS}\ & = \sin \frac{11\alpha}{4} \sin \frac{\alpha}{4} + \sin \frac{7\alpha}{4} \sin \frac{3\alpha}{4} \\[br]\ & = \frac{1}{2}\left[2\sin \frac{11\alpha}{4} \sin \frac{\alpha}{4} + 2\sin \frac{7\alpha}{4} \sin \frac{3\alpha}{4}\right]\\ [br]\ & = \left[ \cos \left( \frac{11\alpha}{4} - \frac{\alpha}{4} \right) - \cos \left( \frac{11\alpha}{4} + \frac{\alpha}{4} \right)+ \cos \left( \frac{7\alpha}{4} - \frac{3\alpha}{4} \right)- \cos \left( \frac{7\alpha}{4} + \frac{3\alpha}{4} \right) \right]\\[br]\ & = \frac{1}{2} \left( \cos \frac{11\alpha-\alpha}{4} - \cos \frac{11\alpha + \alpha}{4} + \cos \frac{7\alpha -3\alpha}{4} -\cos \frac{7\alpha +3\alpha}{4} \right)\\[br]\ & = \frac{1}{2} \left( \cos \frac{10\alpha}{4} -\cos \frac{12\alpha}{4} + \cos \frac{4\alpha}{4} - \cos \frac{10\alpha}{4} \right)\\[br]\ & = \frac{1}{2} \left( \cos \alpha - \cos 3\alpha \right)\\[br]\ & = \frac{1}{2}\times 2 \sin \frac{\alpha +3 \alpha}{2} \sin \frac{3\alpha - \alpha}{2} \\[br]\ & = \sin \frac{4\alpha}{2} \sin \frac{2\alpha}{2}\\[br]\ & = \sin 2\alpha \sin \alpha\\[br]\ & = \text{RHS}[br]\end{align} [/math][br][br]7. (g) [math]\frac{ \cos 10^{\circ} - \sin 10^{\circ} }{ \cos 10^{\circ} + \sin 10^{\circ} = \tan 35^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{ \cos 10^{\circ} - \sin 10^{\circ} }{ \cos 10^{\circ} + \sin 10^{\circ} }\\[br]\ & = \frac{ \sin (90^{\circ}- 10^{\circ}) - \sin 10^{\circ} }{ \sin( 90^{\circ} - 10^{\circ}) + \sin 10^{\circ} }\\[br]\ & = \frac{ \sin 80^{\circ} - \sin 10^{\circ} }{ \sin 80^{\circ} + \sin 10^{\circ} }\\[br]\ & = \frac{2\cos \frac{80^{\circ}+10^{\circ} }{2} \sin \frac{80^{\circ} - 10^{\circ} }{2} }{ 2\sin \frac{80^{\circ}+10^{\circ} }{2} \cos \frac{80^{\circ} - 10^{\circ} }{2} }\\[br]\ & = \frac{\cos 45^{\circ} \sin 35^{\circ}}{\sin 45^{\circ} \cos 35^{\circ}}\\[br]\ & = \frac{ \frac{1}{\sqrt {2}} \sin 35^{\circ} }{ \frac{1}{\sqrt {2}} \cos 35^{\circ} }\\[br]\ & = \frac{ \sin 35^{\circ}}{\cos 35^{\circ} }\\[br]\ & = \tan 35^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]7. (h) [math] \frac{ \cos 75^{\circ} + \cos 15^{\circ} }{ \sin 75^{\circ} - \sin 15^{\circ} } = \sqrt{3} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{ \cos 75^{\circ} + \cos 15^{\circ} }{ \sin 75^{\circ} - \sin 15^{\circ} }\\[br]\ & = \frac{ 2 \cos \frac{75^{\circ}+ 15^{\circ}}{2} \cos \frac{75^{\circ}- 15^{\circ}}{2} }{2 \cos \frac{75^{\circ}+ 15^{\circ}}{2} \sin \frac{75^{\circ} - 15^{\circ}}{2} }\\[br]\ & = \frac{\cos \frac{60^{\circ}}{2}}{\sin \frac{60^{\circ}}{2}}\\[br]\ & = \frac{ \cos 30^{\circ} }{ \sin 30^{\circ} } \\[br]\ & = \cot 30^{\circ} \\[br]\ & = \sqrt{3}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]8. (a) [math] \sin 75^{\circ} + \sin 75^{\circ} = \sqrt{\frac{3}{2} } [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 75^{\circ} + \sin 75^{\circ} \\[br]\ & = 2\sin \frac{ 75^{\circ} +15^{\circ} }{2}\cos \frac{ 75^{\circ} -15^{\circ} }{2}\\[br]\ & = 2\sin \frac {90^{\circ}}{2} \cos \frac{60^{\circ}}{2}\\[br]\ & = 2\sin 45^{\circ} \cos 30^{\circ}\\[br]\ & = 2\times \frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}\\[br]\ & = \frac{ \sqrt{3}}{\sqrt{2}}\\[br]\ & = \sqrt{\frac{3}{2} }\\[br]\ & = \text{RHS}[br]\end{align} [/math][br][br]8. (b) [math] \sin 50^{\circ} + \sin 70^{\circ} = \sqrt{3} \cos 10^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 50^{\circ} + \sin 70^{\circ} \\[br]\ & = 2\sin \frac{ 50^{\circ} +70^{\circ} }{2}\cos \frac{ 50^{\circ} -70^{\circ} }{2}\\[br]\ & = 2\sin \frac {120^{\circ}}{2} \cos \frac{-20^{\circ}}{2}\\[br]\ & = 2\sin 60^{\circ} \cos (-10^{\circ})\\[br]\ & = 2\times \frac{\sqrt{3}}{2}\times \cos 10^{\circ}\\[br]\ & = \sqrt{3} \cos 10^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]8. (c) Prove that: [math] \cos 40^{\circ} + \sin 40^{\circ} = \sqrt{2} \cos 5^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 40^{\circ} + \sin 40^{\circ} \\[br]\ & = \sin (90^{\circ}-40^{\circ}) + \sin 40^{\circ} \\[br]\ & = \sin 50^{\circ} + \sin 40^{\circ} \\[br]\ & = 2\sin \frac{ 50^{\circ} +40^{\circ} }{2}\cos \frac{ 50^{\circ} -40^{\circ} }{2}\\[br]\ & = 2\sin \frac {90^{\circ}}{2} \cos \frac{10^{\circ}}{2}\\[br]\ & = 2\sin 45^{\circ} \cos (5^{\circ})\\[br]\ & = 2\times \frac{1}{\sqrt{2}}\times \cos 5^{\circ}\\[br]\ & = \left(\sqrt{2}\right)^2 \times \frac{1} { \sqrt{2}} \cos 5^{\circ}\\[br]\ & = \sqrt{2} \cos 5^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]8. (d) Prove that: [math] \cos 40^{\circ} - \sin 40^{\circ} = \sqrt{2} \sin 5^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 40^{\circ} - \sin 40^{\circ} \\[br]\ & = \sin (90^{\circ}-40^{\circ}) - \sin 40^{\circ} \\[br]\ & = \sin 50^{\circ} - \sin 40^{\circ} \\[br]\ & = 2\cos \frac{ 50^{\circ} +40^{\circ} }{2}\sin \frac{ 50^{\circ} -40^{\circ} }{2}\\[br]\ & = 2\cos \frac {90^{\circ}}{2} \sin \frac{10^{\circ}}{2}\\[br]\ & = 2\cos 45^{\circ} \sin (5^{\circ})\\[br]\ & = 2\times \frac{1}{\sqrt{2}}\times \sin 5^{\circ}\\[br]\ & = \left(\sqrt{2}\right)^2 \times \frac{1} { \sqrt{2}} \sin 5^{\circ}\\[br]\ & = \sqrt{2} \sin 5^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]8. (e) Prove that: [math]\sin 10^{\circ} + \cos 40^{\circ} = \cos 20^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 10^{\circ} + \cos 40^{\circ} \\[br]\ & = \sin 10^{\circ} + \sin (90^{\circ}- 40^{\circ}) \\[br]\ & = \sin 10^{\circ} + \sin (50^{\circ}) \\[br]\ & = 2 \sin \frac {10^{\circ}+50^{\circ}}{2}\cos \frac {10^{\circ}-50^{\circ}}{2}\\[br]\ & = 2\sin \frac{60^{\circ}}{2}\cos \frac{-40^{\circ}}{2}\\[br]\ & = 2\sin 30^{\circ} \cos 20 ^{\circ}\\[br]\ & = 2\times \frac{1}{2} \cos 20^{\circ}\\[br]\ & = \cos 20^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]8. (f) Prove that: [math]\cos 56^{\circ} + \sin 86^{\circ}= \sqrt{3} \cos 26^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 56^{\circ} + \sin 86^{\circ} \\[br]\ & = \sin(90^{\circ}-56^{\circ}) + \sin 86^{\circ} \\[br]\ & = \sin 34^{\circ} + \sin 86^{\circ} \\[br]\ & = 2\sin \frac{34^{\circ}+86^{\circ}}{2} \cos \frac{34^{\circ}-86^{\circ}}{2}\\[br]\ & = 2\sin \frac{120^{\circ}}{2} \cos \frac{-52}{2}\\[br]\ & =2 \sin 60^{\circ} \cos 26^{\circ}\\[br]\ & = 2\times \sqrt{3}{2} \times \cos 26^{\circ}\\[br]\ & = \sqrt{3} \cos 26^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]9. (a) Prove that: [math]2\cos(45^{\circ} + \theta). \cos (45^{\circ}-\theta) = 2\cos 2\theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = 2\cos(45^{\circ} + \theta). \cos (45^{\circ}-\theta)\\[br]\ & = \cos \left[( 45^{\circ} + \theta) + (45^{\circ}-\theta) \right] + \cos \left[ 45^{\circ} + \theta - (45^{\circ}-\theta) \right]\\[br]\ & = \cos ( 45^{\circ} + \theta + 45^{\circ}-\theta) + \cos (45^{\circ} + \theta - 45^{\circ}+\theta) \\[br]\ & = \cos 90^{\circ} + \cos 2\theta \\[br]\ & = 0 + 2\cos 2\theta\\[br]\ & = 2\cos 2\theta\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]10. (d) Prove that: [math] \frac{\sin 5\theta + \sin 2\theta -\sin \theta}{\cos 5\theta + \cos 2\theta + \cos \theta} = \tan 2\theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\sin 5\theta + \sin 2\theta -\sin \theta}{\cos 5\theta + \cos 2\theta + \cos \theta}\\[br]\ & = \frac{\sin 5\theta -\sin \theta+\sin 2\theta }{\cos 5\theta+ \cos \theta + \cos 2\theta }\\[br]\ & = \frac{2\cos \frac{ 5\theta + \theta}{2} \sin \frac{5\theta -\theta}{2} + \sin 2\theta}{2\cos \frac{5\theta+\theta}{2} \cos \frac{5\theta- \theta}{2} + \cos 2\theta }\\[br]\ & = \frac{ 2\cos \frac{6\theta}{2} \sin \frac{4\theta}{2} + \sin 2\theta}{ 2\cos \frac{6\theta}{2} \sin \frac{4\theta}{2} + \cos 2\theta }\\[br]\ & = \frac{2\cos 3\theta \sin 2\theta + \sin 2\theta}{2\cos 3\theta\cos 2\theta + \cos 2\theta}\\[br]\ & = \frac{ \sin 2\theta ( 2\cos 3\theta + 1) }{ \cos 2\theta(2\cos 3\theta +1) }\\[br]\ & = \frac{\sin 2\theta}{\cos 2\theta}\\[br]\ & = \tan 2\theta \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]10. (e) Prove that: [math] \frac{\cos \theta - \cos 2\theta + \cos 3\theta}{\sin \theta - \sin 2\theta + \sin 3\theta } = \cot 2\theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & =\frac{\cos \theta - \cos 2\theta + \cos 3\theta}{\sin \theta - \sin 2\theta + \sin 3\theta } \\[br]\ & = \frac{\cos 3\theta + \cos \theta - \cos 2\theta}{\sin 3\theta + \sin \theta -\sin 2\theta}\\[br]\ & = \frac{ 2\cos \frac{3\theta + \theta}{2} \cos \frac{3\theta -\theta}{2} - \cos 2\theta}{ 2\sin \frac{3\theta+\theta}{2} \cos \frac{3\theta - \theta}{2} - \sin 2\theta}\\[br]\ & = \frac{2\cos \frac{4\theta}{2} \cos \frac{2\theta}{2} - \cos 2\theta} { 2\sin \frac{4\theta}{2} \cos \frac{2\theta}{2} -\sin 2\theta}\\[br]\ & = \frac{2\cos 2\theta \cos \theta -\cos 2\theta}{2\sin 2\theta \cos \theta - \sin 2\theta}\\[br]\ & = \frac {\cos 2\theta ( 2\cos \theta - 1)}{\sin 2\theta (2\cos \theta -1)}\\[br]\ & = \frac{\cos 2\theta}{\sin 2\theta}\\[br]\ & = \cot 2\theta\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]10. (f) Prove that: [math] \frac{\sin 25^{\circ} +\sin 20^{\circ}+\sin 10^{\circ} + \sin 5^{\circ}}{\cos 5^{\circ}+\cos 10^{\circ} + \cos 20^{\circ}+\cos 25^{\circ}} = \tan 15^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\sin 25^{\circ} +\sin 20^{\circ}+\sin 10^{\circ} + \sin 5^{\circ}}{\cos 5^{\circ}+\cos 10^{\circ} + \cos 20^{\circ}+\cos 25^{\circ}}\\[br]\ & = \frac{\sin 25^{\circ}+\sin 5^{\circ}+\sin 20^{\circ}+\sin 10^{\circ} }{\cos 25^{\circ}+\cos 5^{\circ}+\cos 20^{\circ}+\cos 10^{\circ}}\\[br]\ & = \frac{ 2\sin \frac{25^{\circ}+5^{\circ}}{2} \cos \frac{25^{\circ}-5^{\circ}}{2} +2\sin \frac{20^{\circ}+10^{\circ}}{2} \cos \frac{20^{\circ}-10^{\circ}}{2} } { 2\cos \frac{25^{\circ}+5^{\circ}}{2} \cos \frac{25^{\circ}-5^{\circ}}{2} +2\cos \frac{20^{\circ}+10^{\circ}}{2} \cos \frac{20^{\circ}-10^{\circ}}{2} }\\[br]\ & = \frac{2\sin \frac{30^{\circ}}{2} \cos \frac{20^{\circ}}{2}+2\sin \frac{30^{\circ}}{2} \cos \frac{10^{\circ}}{2} }{2\cos \frac{30^{\circ}}{2} \cos \frac{20^{\circ}}{2}+2\cos \frac{30^{\circ}}{2} \cos \frac{10^{\circ}}{2}}\\[br]\ & = \frac{2\sin 15 ^{\circ} \cos 10^{\circ} + 2\sin 15^{\circ} \cos 5^{\circ}}{2\cos 15 ^{\circ} \cos 10^{\circ} + 2\cos 15^{\circ} \cos 5^{\circ} }\\[br]\ & = \frac{2\sin 15^{\circ} ( \cos 10^{\circ}+\cos 5^{\circ})}{2\cos15^{\circ}( \cos 10^{\circ} + \cos 5^{\circ})}\\[br]\ & = \frac{\sin 15^{\circ}}{\cos 15^{\circ}}\\[br]\ & = \tan 15^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]10. (g) Prove that: [math]\frac{1-\cos10^{\circ} + \cos 40^{\circ}-\cos 50^{\circ}}{1+\cos 10^{\circ}-\cos 40^{\circ}-\cos 50^{\circ}}=\cot 20^{\circ} \tan 5^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{1-\cos10^{\circ} + \cos 40^{\circ}-\cos 50^{\circ}}{1+\cos 10^{\circ}-\cos 40^{\circ}-\cos 50^{\circ}}\\[br]\ & = \frac{ 1 - (1-2\sin^2 5^{\circ}) + (\cos 40^{\circ}-\cos 50^{\circ})}{ 1+ (2\cos^2 5^{\circ} -1) - (\cos 40^{\circ} + \cos 50^{\circ})}\\[br]\ & = \frac{1-1+2\sin^2 5^{\circ}+2\sin \frac{40^{\circ}+50^{\circ}}{2} \sin \frac{ 50^{\circ}-40^{\circ}}{2}}{ 1+2\cos^2 5^{\circ}-1-2\cos \frac{40^{\circ}+50^{\circ}}{2} \cos \frac{40^{\circ}-50^{\circ}}{2}}\\[br]\ & = \frac{2\sin^2 5^{\circ}+2\sin 45^{\circ} \sin 5^{\circ}}{2\cos^2 5^{\circ}-2\cos 45^{\circ}\cos 5^{\circ}}\\[br]\ & = \frac{ 2\sin 5^{\circ} (\sin 5^{\circ} + \sin 45^{\circ})}{2\cos 5^{\circ}(\cos 5^{\circ}-\cos 45^{\circ})}\\[br]\ & = \frac{\sin 5^{\circ}}{\cos 5^{\circ}} \times \frac{\sin 5^{\circ}+\sin 45^{\circ}}{\cos 5^{\circ} - \cos 45^{\circ}}\\[br]\ & = \tan 5^{\circ} \times \frac{2\sin \frac{5^{\circ}+45^{\circ}}{2} \cos \frac{5^{\circ}-45^{\circ}}{2}}{2\sin \frac{5^{\circ}+45^{\circ}}{2} \sin \frac{45^{\circ}-5^{\circ}}{2}}\\[br]\ & = \tan 5^{\circ} \frac{\cos 20^{\circ}}{\sin 20^{\circ}}\\[br]\ & = \tan 5^{\circ} \cot 20^{\circ}\\[br]\ & = \cot 20^{\circ} \tan 5^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]11. (a) Prove that: [math] \frac{\sin 2A \sin A + \sin 6A \sin 3A}{ \cos 2A \sin A + \cos 6A \sin 3A}=\tan 5A [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\sin 2A \sin A + \sin 6A \sin 3A}{ \cos 2A \sin A + \cos 6A \sin 3A}\\[br]\ & = \frac{2\sin 2A \sin A + 2\sin 6A \sin 3A}{ 2\cos 2A \sin A + 2\cos 6A \sin 3A}\\[br]\ & = \frac{\cos (2A-A)-\cos (2A+A) + \cos (6A-3A)-\cos (6A+3A)}{\sin (2A+A) -\sin (2A-A) + \sin (6A+3A) - \sin (6A-3A) }\\[br]\ & = \frac{\cos A -\cos 3A + \cos 3A - \cos 9A}{\sin 3A - \sin A +\sin 9A-\sin 3A}\\[br]\ & = \frac{\cos A - \cos 9A}{\sin 9A - \sin A}\\[br]\ & = \frac{2\sin \frac{A+9A}{2} \sin \frac{9A-A}{2}}{2\cos \frac{9A+A}{2}\sin \frac{9A-A}{2}}\\[br]\ & = \frac{\sin 5A}{\cos 5A}\\[br]\ & = \tan 5A\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]11. (b) Prove that: [math] -\tan \theta = \frac{\cos 5\theta .\sin 2\theta - \cos 4\theta . \sin 3\theta}{\sin 5\theta \sin 2\theta + \cos 4\theta \cos 3\theta }[/math][br]Solution:[br][math] \begin{align}[br]\text{RHS} \ & = \frac{\cos 5\theta .\sin 2\theta - \cos 4\theta . \sin 3\theta}{\sin 5\theta \sin 2\theta ‌‌‍+ \cos 4\theta \cos 3\theta }\\[br]\ & = \frac{2\cos 5\theta .\sin 2\theta - 2\cos 4\theta . \sin 3\theta}{2\sin 5\theta \sin 2\theta + 2\cos 4\theta \cos 3\theta }\\[br]\ & = \frac{\sin (5\theta +2\theta) - \sin (5\theta - 2\theta) - \{ \sin (4\theta +3\theta) - \sin ( 4\theta -3\theta) \} }{\cos (5\theta - 2\theta) - \cos (5\theta +2\theta) + \cos (4\theta+3\theta) + \cos (4\theta -3\theta) }\\[br]\ & = \frac{\sin 7\theta - \sin 3\theta - ( \sin 7\theta -\sin \theta )}{\cos 3\theta - \cos 7 \theta + \cos 7\theta + \cos \theta)}\\[br]\ & = \frac{\sin 7\theta - \sin 3\theta - \sin 7\theta + \sin \theta }{\cos 3\theta - \cos 7 \theta + \cos 7\theta - \cos \theta}\\[br]\ & = \frac{ - \sin 3\theta +\sin \theta }{ \cos 3\theta + \cos \theta }\\[br]\ & = \frac{ \sin \theta - \sin 3\theta}{ \cos 3\theta + \cos \theta }\\[br]\ & = \frac{ 2\cos \frac{3\theta +\theta}{2} \sin \frac{\theta -3\theta}{2}}{2\cos \frac{\theta +3\theta}{2} \cos \frac{3\theta -\theta}{2}}\\[br]\ & = \frac{ \sin (-\theta)}{cos \theta }\\[br]\ & = -\frac{\sin \theta}{cos \theta} \\[br]\ & = -\tan \theta \\[br]\ & = \text{LHS}\\[br]\end{align} [/math][br][br]11. (c) Prove that: [math] \frac{ \sin 18^{\circ} \cos 24^{\circ}-\cos 12^{\circ} \sin 6^{\circ} }{ \sin 24^{\circ}\sin 6^{\circ} + \cos 36^{\circ} \cos 6^{\circ} } = \tan 12^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{ \sin 18^{\circ} \cos 24^{\circ}-\cos 12^{\circ} \sin 6^{\circ} }{ \sin 24^{\circ}\sin 6^{\circ} + \cos 36^{\circ} \cos 6^{\circ} } \\[br] \ & = \frac{ 2\sin 18^{\circ} \cos 24^{\circ}-2\cos 12^{\circ} \sin 6^{\circ} }{ 2\sin 24^{\circ}\sin 6^{\circ} + 2\cos 36^{\circ} \cos 6^{\circ} } \\[br]\ & = \frac{\sin( 18^{\circ}+24^{\circ}) +\sin (18^{\circ}-24^{\circ}) - \left\{ \sin (12^{\circ} +6^{\circ}) - \sin ( 12^{\circ}-6^{\circ}) \right\}}{\cos ( 24^{\circ}-6^{\circ}) - \cos ( 24^{\circ}+6^{\circ}) + \cos (36^{\circ}+6^{\circ}) + \cos (36^{\circ}-6^{\circ})}\\[br]\ & = \frac{\sin 42^{\circ}+\sin (-6^{\circ}) - \sin 18^{\circ} + \sin 6^{\circ}}{\cos 18^{\circ} - \cos 30^{\circ}+\cos 42^{\circ} +\cos 30^{\circ}}\\[br]\ & = \frac{\sin 42^{\circ}-\sin 6^{\circ} - \sin 18^{\circ} + \sin 6^{\circ}}{\cos 18^{\circ} +\cos 42^{\circ} }\\[br]\ & = \frac{\sin 42^{\circ} - \sin 18^{\circ} }{\cos 18^{\circ} +\cos 42^{\circ} }\\[br]\ & = \frac{2\cos \frac{42^{\circ}+18^{\circ}}{2} \sin \frac{42^{\circ}-18^{\circ}}{2} }{2\cos \frac{18^{\circ}+42^{\circ}}{2} \cos \frac{18^{\circ}-42^{\circ}}{2} }\\[br]\ & = \frac{2\cos \frac{60^{\circ}}{2} \sin \frac{24^{\circ}}{2} }{2\cos \frac{60^{\circ}}{2} \cos \frac{-24}{2} }\\[br]\ & = \frac{2\cos 30^{\circ} \sin 12^{\circ}}{2\cos 30^{\circ} \cos 12^{\circ}}\\[br]\ & = \frac{\sin 12^{\circ}}{\cos 12^{\circ}}\\[br]\ & = \tan 12^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]11. (d) Prove that: [math] \frac{ \sin 5^{\circ} \cos 10^{\circ}+\sin 15^{\circ} \cos 30^{\circ} }{ \sin 5^{\circ}\sin 10^{\circ} + \sin 15^{\circ} \sin 30^{\circ} } = \cot 25^{\circ} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{ \sin 5^{\circ} \cos 10^{\circ}+\sin 15^{\circ} \cos 30^{\circ} }{ \sin 5^{\circ}\sin 10^{\circ} + \sin 15^{\circ} \sin 30^{\circ} } \\[br]\ & = \frac{2\cos 10^{\circ} \sin 5^{\circ} + 2\cos 30^{\circ}\sin 15^{\circ}}{2\sin 10^{\circ} \sin 5^{\circ} + 2\sin 30^{\circ}\sin 15^{\circ}}\\[br]\ & = \frac{\sin(10^{\circ}+5^{\circ}) - \sin (10^{\circ}-5^{\circ}) + \{ \sin (30^{\circ}+15^{\circ}) - \sin (30^{\circ}-15^{\circ}) \}}{\cos(10^{\circ}-5^{\circ}) - \cos (10^{\circ}+5^{\circ}) + \{ \cos (30^{\circ}-15^{\circ}) - \cos (30^{\circ}+15^{\circ}) \}}\\[br]\ & = \frac{\sin 15^{\circ} -\sin 5^{\circ} + \sin 45^{\circ} - \sin 15^{\circ} }{\cos 5^{\circ} -\cos 15^{\circ} + \cos 15^{\circ} - \cos 45^{\circ}}\\[br]\ & = \frac{ \sin 45^{\circ} -\sin 5^{\circ}}{\cos 5^{\circ} -\cos 45^{\circ}}\\[br]\ & = \frac{2\cos \frac{45^{\circ} + 5^{\circ}}{2} \sin \frac{45^{\circ}-5^{\circ}}{2}}{2\sin \frac{5^{\circ}+45^{\circ}}{2} \sin \frac{45^{\circ}-5^{\circ}}{2}}\\[br]\ & =\frac{\cos\frac{50^{\circ}}{2} }{\sin \frac{50^{\circ}}{2} }\\[br]\ & =\frac{\cos 25^{\circ}}{\sin 25^{\circ}}\\[br]\ & =\cot 25^{\circ}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]12. (a) Prove that: [math]\cos \theta . \cos (60^{\circ}-\theta)\cos(60^{\circ}+\theta) = \frac{1}{4} \cos 3\theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos \theta . \cos (60^{\circ}-\theta)\cos(60^{\circ}+\theta) \\[br]\ & = \frac{1}{2} \cos \theta [ 2\cos (60^{\circ}+\theta)\cos(60^{\circ}-\theta) ]\\[br]\ & = \frac{1}{2} \cos \theta [\cos ( 60^{\circ} + \theta + 60^{\circ} -\theta)+ \cos (60^{\circ}+\theta -60^{\circ}+\theta) ]\\[br]\ & = \frac{1}{2}\cos \theta [\cos 120^{\circ} +\cos 2\theta ]\\[br]\ & = \frac{1}{2}\cos \theta [\frac{- 1}{2} + \cos 2\theta ]\\[br]\ & = \frac{-1}{4} \cos \theta + \frac{1}{2} \cos 2\theta \cos \theta \\[br]\ & = \frac{-1}{4} \cos\theta +\frac{1}{4}[2\cos 2\theta \cos \theta ]\\[br]\ & = \frac{-1}{4} \cos \theta + \frac{1}{4} [ \cos (2\theta +\theta) + \cos (2\theta -\theta) ]\\[br]\ & = \frac{-1}{4} \cos \theta + \frac{1}{4} [ \cos (2\theta+\theta) + \cos (2\theta -\theta) ]\\[br]\ & = \frac{-1}{4}\cos \theta + \frac{1}{4} [\cos 3\theta + \cos \theta ]\\[br]\ & = \frac{-1}{4} \cos \theta + \frac{1}{4} \cos 3\theta +\frac{1}{4} \cos \theta\\[br]\ & = \frac{1}{4} \cos 3\theta \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]12. (b) Prove that: [math]\sec \left( \frac{\pi^c}{4} + \frac{\alpha}{2} \right) \times \sec \left( \frac{\pi^c}{4} - \frac{\alpha}{2} \right) = 2\sec \alpha [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sec \left( \frac{\pi^c}{4} + \frac{\alpha}{2} \right) \times \sec \left( \frac{\pi^c}{4} - \frac{\alpha}{2} \right) \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]12. (c) Prove that: [math] \text{cosec}(45^{\circ}+\theta)\text{cosec}( 45^{\circ}-\theta) = 2\sec 2\theta [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \text{cosec}(45^{\circ}+\theta)\text{cosec}( 45^{\circ}-\theta) \\[br]\ & = \frac{1}{\sin(45^{\circ}+\theta) \sin (45^{\circ}-\theta)}\\[br]\ & = \frac{2}{2\sin(45^{\circ}+\theta) \sin (45^{\circ}-\theta)}\\[br]\ & = \frac{2}{\cos \{(45^{\circ}+\theta) - (45^{\circ}-\theta) \} - \cos \{(45^{\circ}+\theta) + (45^{\circ}-\theta) \} }\\[br]\ & = \frac{2}{\cos (45^{\circ} + \theta - 45^{\circ}+ \theta) - \cos ( 45^{\circ}+\theta + 45^{\circ}-\theta) }\\[br]\ & = \frac{2}{\cos 2\theta - \cos 90^{\circ} }\\[br]\ & = \frac{2}{\cos 2\theta - 0 }\\[br]\ & = \frac{2}{\cos 2\theta}\\[br]\ & = 2\sec 2\theta \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]12. (d) Prove that: [math] \cos^2 \alpha + \cos^2(\alpha +120^{\circ}) + \cos^2 (\alpha - 120^{\circ}) = \frac{3}{2} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos^2 \alpha + \cos^2(\alpha +120^{\circ}) + \cos^2 (\alpha - 120^{\circ}) \\[br]\ & = \frac{1+\cos 2\alpha }{2} + \frac{1+\cos2(\alpha +120^{\circ})}{2}+ \frac{1+\cos 2(\alpha - 120^{\circ})}{2}\\[br]\ & = \frac{1+\cos 2\alpha }{2} + \frac{1+\cos(2\alpha +240^{\circ})}{2}+ \frac{1+\cos (2\alpha - 240^{\circ})}{2}\\[br]\ & = \frac{1+\cos2\alpha + 1+\cos (2\alpha + 240^{\circ})+ 1+\cos (2\alpha - 240^{\circ}) }{2}\\[br]\ & = \frac{3+ \cos 2\alpha + [\cos (2\alpha + 240^{\circ})+\cos (2\alpha - 240^{\circ})] }{2}\\[br]\ & = \frac{3+ \cos 2\alpha + 2\cos 2\alpha \cos 240^{\circ}}{2}\\[br]\ & = \frac{3+\cos 2\alpha +2\cos 2\alpha \times \left(\frac{-1}{2}\right)}{2}\\[br]\ & = \frac{3+\cos 2\alpha - \cos 2\alpha}{2}\\[br]\ & = \frac{3}{2}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (a) Prove that: [math] \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} =\frac{1}{8} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} \\[br]\ & = \frac{1}{2}\sin 10^{\circ} [ 2\sin 70^{\circ} \sin 50^{\circ} ]\\[br]\ & = \frac{1}{2} \sin 10^{\circ} [ \cos (70^{\circ}-50^{\circ}) - \cos ( 70^{\circ} + 50^{\circ}] \\[br]\ & = \frac{1}{2} \sin 10^{\circ} [ \cos 20^{\circ}-\cos 120^{\circ}]\\[br]\ & = \frac{1}{2} \sin 10^{\circ} \left[ \cos 20^{\circ} + \frac{1}{2} \right] \\[br]\ & = \frac{1}{2} \cos 20^{\circ}\sin 10^{\circ} + \frac{1}{4} \sin 10^{\circ} \\[br]\ & = \frac{1}{4}[2 \cos 20^{\circ}\sin 10^{\circ}] + \frac{1}{4} \sin 10^{\circ} \\[br]\ & = \frac{1}{4}[\sin (20^{\circ}+10^{\circ}) - \sin (20^{\circ}-10^{\circ})] + \frac{1}{4} \sin 10^{\circ} \\[br]\ & = \frac{1}{4}[\sin 30^{\circ}- \sin 10^{\circ}] + \frac{1}{4} \sin 10^{\circ} \\[br]\ & = \frac{1}{4}\left[\frac{1}{2}- \sin 10^{\circ}\right] + \frac{1}{4} \sin 10^{\circ} \\[br]\ & = \frac{1}{8} - \frac{1}{4} \sin 10^{\circ} + \frac{1}{4} \sin 10^{\circ}\\[br]\ & = \frac{1}{8}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (b) Prove that: [math] 8\cos 20^{\circ}\cos40^{\circ}\cos80^{\circ}=1 [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = 8\cos 20^{\circ}\cos40^{\circ}\cos80^{\circ} \\[br]\ & = 4 \cos 20^{\circ} [2\cos 80^{\circ} \cos 40^{\circ} ]\\[br]\ & = 4\cos 20^{\circ} [ \cos (80^{\circ} +40^{\circ}) + \cos (80^{\circ}-40^{\circ}) ]\\[br]\ & = 4\cos 20^{\circ}[ \cos 120^{\circ} + \cos 40^{\circ} ]\\[br]\ & = 4\cos 20^{\circ} \left[ \frac{-1}{2} + \cos 40^{\circ} \right]\\[br]\ & = -2\cos 20^{\circ} + 4\cos 40^{\circ} \cos 20^{\circ}\\[br]\ & = -2\cos 20^{\circ}+ 2[2\cos 40^{\circ}\cos 20^{\circ} ]\\[br]\ & = -2\cos 20^{\circ} +2[\cos(40^{\circ}+20^{\circ}) +\cos (40^{\circ}-20^{\circ})]\\[br]\ & = -2\cos 20^{\circ}+2[\cos 60^{\circ}+\cos 20^{\circ}]\\[br]\ & = -2\cos 20^{\circ} + 2\cos 60^{\circ}+2\cos 20^{\circ}\\[br]\ & = 2\cos 60^{\circ}\\[br]\ & = 2\times \frac{1}{2}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (c) Prove that: [math] \cos40^{\circ}\cos80^{\circ}\cos 160^{\circ}= -\frac{1}{8} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos40^{\circ}\cos80^{\circ}\cos 160^{\circ} \\[br]\ & = \frac{1}{2} \cos 40^{\circ} [ 2\cos 160^{\circ}\cos 80^{\circ}]\\[br]\ & = \frac{1}{2} \cos 40^{\circ}[ \cos (160^{\circ}+80^{\circ}) - \cos (160^{\circ}-80^{\circ}) ]\\[br]\ & = \frac{1}{2} \cos 40^{\circ} [\cos 240^{\circ}-\cos 80^{\circ} ]\\[br]\ & = \frac{1}{2} \cos 40^{\circ} \left[ \frac{-1}{2} -\cos 80^{\circ} \right]\\[br]\ & = \frac{-1}{4} \cos 40^{\circ} - \frac{1}{2} \cos 80^{\circ}\cos 40^{\circ}\\[br]\ & = \frac{-1}{4} \cos 40^{\circ} -\frac{1}{4} [2\cos 80^{\circ} \cos 40^{\circ} ]\\[br]\ & = \frac{-1}{4} \cos 40^{\circ} -\frac{1}{4} [\cos (80^{\circ}+40^{\circ}) -\cos (80^{\circ}-40^{\circ}) ]\\[br]\ & = \frac{-1}{4} \cos 40^{\circ} -\frac{1}{4} [\cos 120^{\circ} -\cos 40^{\circ} ]\\[br]\ & = \frac{-1}{4} \cos 40^{\circ} -\frac{1}{4} \cos 120^{\circ} + \frac{1}{4}\cos 40^{\circ}\\[br]\ & = -\frac{1}{4} \cos 120^{\circ} \\[br]\ & = -\frac{1}{4} \times \frac{1}{2} \\[br]\ & = -\frac{1}{8}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (d) Prove that: [math] \sin 20^{\circ} \sin 30^{\circ}\sin 40^{\circ}\sin 80^{\circ}= \frac{\sqrt{3}}{16} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 20^{\circ} \sin 30^{\circ}\sin 40^{\circ}\sin 80^{\circ} \\[br]\ & = \sin 20^{\circ} \times \frac{1}{2} \times \sin 40^{\circ}\sin 80^{\circ} \\[br]\ & = \frac{1}{2}\sin 20^{\circ} \sin 40^{\circ}\sin 80^{\circ} \\[br]\ & = \frac{1}{4}\sin 20^{\circ}[2 \sin 80^{\circ}\sin 40^{\circ}] \\[br]\ & = \frac{1}{4}\sin 20^{\circ}[\cos ( 80^{\circ}-40^{\circ}) - \cos ( 80^{\circ}+40^{\circ})] \\[br]\ & = \frac{1}{4}\sin 20^{\circ}[\cos 40^{\circ}) - \cos 120^{\circ}] \\[br]\ & = \frac{1}{4}\sin 20^{\circ}\left[\cos 40^{\circ} - \left(\frac{-1}{2}\right)\right] \\[br]\ & = \frac{1}{4} \cos 40^{\circ} \sin 20^{\circ} + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{1}{8}[2 \cos 40^{\circ} \sin 20^{\circ}] + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{1}{8}[\sin (40^{\circ}+20^{\circ}) - \sin (40^{\circ}-20^{\circ}) ] + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{1}{8}[\sin 60^{\circ}- \sin 20^{\circ} ] + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{1}{8}\left[\frac{\sqrt{3}}{2}- \sin 20^{\circ} \right] + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{\sqrt{3}}{16}- \frac{1}{8} \sin 20^{\circ} + \frac{1}{8} \sin 20^{\circ}\\[br]\ & = \frac{\sqrt{3}}{16}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (e) Prove that: [math] \sin 10^{\circ} \sin 50^{\circ}\sin 60^{\circ}\sin 70^{\circ}= \frac{\sqrt{3}}{16} [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \sin 10^{\circ} \sin 50^{\circ}\sin 60^{\circ}\sin 70^{\circ} \\[br] \ & = \sin 10^{\circ} \sin 50^{\circ}\times \frac{\sqrt{3}}{2} \times \sin 70^{\circ} \\[br] \ & = \frac{\sqrt{3}}{2}\sin 10^{\circ} \sin 50^{\circ}\sin 70^{\circ} \\[br] \ & = \frac{\sqrt{3}}{4}\sin 10^{\circ} [2 \sin 70^{\circ}\sin 50^{\circ} ] \\[br]\ & =\frac{\sqrt{3}}{4} \sin 10^{\circ} [ \cos (70^{\circ}-50^{\circ}) - \cos ( 70^{\circ} + 50^{\circ}] \\[br]\ & = \frac{\sqrt{3}}{4} \sin 10^{\circ} [ \cos 20^{\circ}-\cos 120^{\circ}]\\[br]\ & = \frac{\sqrt{3}}{4} \sin 10^{\circ} \left[ \cos 20^{\circ} + \frac{1}{2} \right] \\[br]\ & = \frac{\sqrt{3}}{4} \cos 20^{\circ}\sin 10^{\circ} +\frac{\sqrt{3}}{8} \sin 10^{\circ} \\[br]\ & = \frac{\sqrt{3}}{8}[2 \cos 20^{\circ}\sin 10^{\circ}] + \frac{\sqrt{3}}{8}\sin 10^{\circ} \\[br]\ & =\frac{\sqrt{3}}{8}[\sin (20^{\circ}+10^{\circ}) - \sin (20^{\circ}-10^{\circ})] + \frac{\sqrt{3}}{8} \sin 10^{\circ} \\[br]\ & =\frac{\sqrt{3}}{8}[\sin 30^{\circ}- \sin 10^{\circ}] +\frac{\sqrt{3}}{8}\sin 10^{\circ} \\[br]\ & = \frac{\sqrt{3}}{8}\left[\frac{1}{2}- \sin 10^{\circ}\right] +\frac{\sqrt{3}}{8} \sin 10^{\circ} \\[br]\ & = \frac{\sqrt{3}}{16} - \frac{\sqrt{3}}{8}\sin 10^{\circ} +\frac{\sqrt{3}}{8} \sin 10^{\circ}\\[br]\ & = \frac{\sqrt{3}}{16}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (f) Prove that: [math] \cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ} =\frac{1}{16} [/math] [br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ} \\[br]\ & = \cos 20^{\circ} \cos 40^{\circ}\times \frac{1}{2} \times \cos 80^{\circ} \\[br]\ & = \frac{1}{2} \cos 20^{\circ} \cos 40^{\circ} cos 80^{\circ} \\[br]\ & = \frac{1}{4} \cos 20^{\circ} [2\cos 80^{\circ} cos 40^{\circ}] \\[br]\ & = \frac{1}{4} \cos 20^{\circ} [\cos (80^{\circ} + 40^{\circ}) + \cos (80^{\circ}-40^{\circ}) ] \\[br]\ & = \frac{1}{4} \cos 20^{\circ} [\cos 120^{\circ} + \cos 40^{\circ} ] \\[br]\ & = \frac{1}{4} \cos 20^{\circ} \left[\frac{-1}{2} + \cos 40^{\circ} \right] \\[br]\ & = \frac{-1}{8} \cos 20^{\circ} + \frac{1}{4}\cos 40^{\circ} \cos 20^{\circ} \\[br]\ & = \frac{-1}{8} \cos 20^{\circ} + \frac{1}{8}[2\cos 40^{\circ} \cos 20^{\circ} ] \\[br]\ & = \frac{-1}{8} \cos 20^{\circ} + \frac{1}{8}[\cos (40^{\circ}+20^{\circ}) + \cos (40^{\circ}-20^{\circ}) ] \\[br]\ & = \frac{-1}{8} \cos 20^{\circ} + \frac{1}{8}[\cos 60^{\circ} + \cos 20^{\circ} ] \\[br]\ & = \frac{-1}{8} \cos 20^{\circ} + \frac{1}{8}\cos 60^{\circ} + \frac{1}{8}\cos 20^{\circ} \\[br]\ & = \frac{1}{8}\cos 60^{\circ}\\[br]\ & = \frac{1}{8} \times \frac{1}{2}\\[br]\ & = \frac{1}{16}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br][br]13. (g) Prove that: [math] \cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ} =\frac{3}{16} [/math] [br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ} \\[br]\ & = \cos 10^{\circ}\times \frac{\sqrt{3}}{2}\times \cos 40^{\circ} \cos 80^{\circ} \\[br]\ & =\frac{\sqrt{3}}{2} \cos 10^{\circ} \cos 70^{\circ} cos 50^{\circ} \\[br]\ & = \frac{\sqrt{3}}{4} \cos 10^{\circ} [2\cos 70^{\circ} cos 50^{\circ}] \\[br]\ & = \frac{\sqrt{3}}{4} \cos 10^{\circ} [\cos (70^{\circ} + 50^{\circ}) + \cos (70^{\circ}-50^{\circ}) ] \\[br]\ & = \frac{\sqrt{3}}{4} \cos 10^{\circ} [\cos 120^{\circ} + \cos 20^{\circ} ] \\[br]\ & = \frac{\sqrt{3}}{4} \cos 10^{\circ} \left[\frac{-1}{2} + \cos 20^{\circ} \right] \\[br]\ & = \frac{-\sqrt{3}}{8} \cos 10^{\circ} + \frac{\sqrt{3}}{4}\cos 20^{\circ} \cos 10^{\circ} \\[br]\ & = \frac{-\sqrt{3}}{8} \cos 10^{\circ} + \frac{\sqrt{3}}{8}[2\cos 20^{\circ} \cos 10^{\circ} ] \\[br]\ & = \frac{-\sqrt{3}}{8} \cos 10^{\circ} + \frac{\sqrt{3}}{8}[\cos (20^{\circ}+10^{\circ}) + \cos (20^{\circ}-10^{\circ}) ] \\[br]\ & = \frac{-\sqrt{3}}{8} \cos 10^{\circ} + \frac{\sqrt{3}}{8}[\cos 30^{\circ} + \cos 10^{\circ} ] \\[br]\ & = \frac{-\sqrt{3}}{8} \cos 10^{\circ} + \frac{\sqrt{3}}{8}\cos 30^{\circ} + \frac{\sqrt{3}}{8}\cos 10^{\circ} \\[br]\ & = \frac{\sqrt{3}}{8}\cos 30^{\circ}\\[br]\ & = \frac{\sqrt{3}}{8} \times \frac{\sqrt{3}}{2}\\[br]\ & = \frac{3}{16}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]13. (h) Prove that: [math] \cos 12^{\circ} \cos 24^{\circ} \cos 48^{\circ} \cos 84^{\circ} =\frac{1}{16} [/math] [br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos 12^{\circ} \cos 24^{\circ} \cos 48^{\circ} \cos 84^{\circ} \\[br]\ & = \frac{(2\sin 12^{\circ} \cos 12^{\circ}) \cos 24^{\circ} \cos 48^{\circ} \cos 84^{\circ}}{2\sin 12^{\circ}} \\[br]\ & = \frac{\sin 24^{\circ} \cos 24^{\circ} \cos 48^{\circ} \cos 84^{\circ}}{2\sin 12^{\circ}} \times \frac{2}{2} \\[br]\ & = \frac{(2\sin 24^{\circ} \cos 24^{\circ}) \cos 48^{\circ} \cos 84^{\circ}}{4\sin 12^{\circ}} \\[br]\ & = \frac{\sin 48^{\circ} \cos 48^{\circ} \cos 84^{\circ}}{4\sin 12^{\circ}}\times \frac{2}{2} \\[br]\ & = \frac{(2\sin 48^{\circ} \cos 48^{\circ} ) \cos 84^{\circ}}{8\sin 12^{\circ}} \\[br]\ & = \frac{\sin 96^{\circ} \cos 84^{\circ}}{8\sin 12^{\circ}} \times \frac{2}{2} \\[br]\ & = \frac{2\sin (90^{\circ}+6^{\circ}) \cos (90^{\circ}-6^{\circ})}{16 \sin 12^{\circ}}\\[br]\ & = \frac{2\cos 6^{\circ} \sin 6^{\circ}}{16 \sin 12^{\circ}}\\[br]\ & = \frac{2\sin 6^{\circ} \cos 6^{\circ}}{16 \sin 12^{\circ}}\\[br]\ & = \frac{\sin 12^{\circ}}{16 \sin 12^{\circ}}\\[br]\ & = \frac{1}{16}\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]14. (a) Prove that: [math] \frac{\sin^2\theta-\sin^2 \alpha}{\sin \theta \cos \theta - \sin \alpha \cos \alpha } =\tan (\theta +\alpha) [/math] [br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\sin^2\theta-\sin^2 \alpha}{\sin \theta \cos \theta - \sin \alpha \cos \alpha } \\[br]\ & = \frac{\sin^2\theta-\sin^2 \alpha}{\sin \theta \cos \theta - \sin \alpha \cos \alpha } \times \frac{2}{2}\\[br]\ & = \frac{2\sin^2\theta-2\sin^2 \alpha}{2\sin \theta \cos \theta -2 \sin \alpha \cos \alpha } \\[br]\ & = \frac{1-\cos 2\theta - (1-\cos 2\alpha) }{\sin 2\theta - \sin 2\alpha}\\[br]\ & = \frac{1-\cos 2\theta - 1+\cos 2\alpha }{\sin 2\theta - \sin 2\alpha}\\[br]\ & = \frac{\cos 2\alpha-\cos 2\theta }{\sin 2\theta - \sin 2\alpha}\\[br]\ & = \frac{2\sin \frac{2\theta + 2\alpha }{2} \sin \frac{2\theta - 2\alpha}{2}}{2\cos \frac{2\theta +2\alpha}{2} \sin \frac{2\theta -2\alpha }{2}}\\[br]\ & = \frac{\sin ( \theta + \alpha)}{\cos (\theta +\alpha) }\\[br]\ & = \tan (\theta +\alpha)\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]14. (b) Prove that: [math] \frac{\cos^2 A-\cos^2 B}{\sin A \cos A - \sin B \cos B } =-\tan (A+B) [/math] [br]Solution:[br][math] \begin{align} [br]\text{LHS} \ & = \frac{\cos^2 A-\cos^2 B}{\sin A \cos A - \sin B \cos B } \\[br]\ & = \frac{\cos^2 A-\cos^2 B}{\sin A \cos A - \sin B \cos B } \times \frac{2}{2} \\[br]\ & = \frac{2\cos^2 A-2\cos^2 B}{2\sin A \cos A - 2\sin B \cos B } \\[br]\ & = \frac{1+\cos 2A -(1+\cos 2B)}{\sin 2A -\sin 2B}\\[br]\ & = \frac{1+\cos 2A -1-\cos 2B}{\sin 2A -\sin 2B}\\[br]\ & = \frac{\cos 2A -\cos 2B}{\sin 2A -\sin 2B}\\[br]\ & = \frac{2\sin \frac{2A+2B}{2} \sin \frac{2B-2A}{2}}{2\cos \frac{2A+2B}{2} \sin \frac{2A-2B}{2}}\\[br]\ & = \frac{\sin (A+B) \sin (B-A) }{\cos (A+B) \sin (A-B)}\\[br]\ & =-\frac{\sin (A+B) \sin (A-B) }{\cos(A+B) \sin (A-B) }\\[br]\ & =-\frac{\sin (A+B) }{\cos(A+B)}\\[br]\ & = -\tan ( A+B) \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]14. (c) Prove that: [math]\frac{\sin 130^{\circ} - \sin 140^{\circ}}{\cos 220^{\circ}+\cos 310^{\circ}} = -1 [/math][br]Solution:[br][math] \begin{align} [br]\text{LHS} \ & = \frac{\sin 130^{\circ} - \sin 140^{\circ}}{\cos 220^{\circ}+\cos 310^{\circ}} \\[br] \ & = \frac{\sin (180^{\circ}-50^{\circ}) - \sin (180^{\circ}-40^{\circ})}{\cos (180^{\circ}+40^{\circ})+\cos(360^{\circ} -50^{\circ})} \\[br]\ & = \frac{\sin 50^{\circ}-\sin 40^{\circ}}{-\cos40^{\circ} + \cos 50^{\circ}}\\[br]\ & = \frac{\sin 50^{\circ}-\sin 40^{\circ}}{ \cos 50^{\circ}-\cos40^{\circ} }\\[br]\ & = \frac{2\cos \frac{50^{\circ}+40^{\circ}}{2} \sin \frac{50^{\circ}-40^{\circ}}{2}}{2\sin \frac{50^{\circ}+40^{\circ}}{2} \sin \frac{40^{\circ}-50^{\circ}}{2}}\\[br]\ & = \frac{\cos 45^{\circ}\sin 5^{\circ}}{\sin 45^{\circ} \sin (-5^{\circ})}\\[br]\ & = -\frac{\frac{1}{\sqrt{2}} \sin 5^{\circ}}{\frac{1}{\sqrt{2}} \sin 5^{\circ}}\\[br]\ & = -1\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]14. (d) Prove that: [math] \frac{\sin^2 \alpha - \sin^2 \beta}{ \sin \alpha \cos \alpha + \sin \beta \cos \beta }= \tan (\alpha -\beta) [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \frac{\sin^2\alpha-\sin^2 \beta}{\sin \alpha \cos \alpha + \sin \beta \cos \beta } \\[br]\ & = \frac{\sin^2\alpha-\sin^2 \beta}{\sin \alpha \cos \alpha + \sin \beta \cos \beta } \times \frac{2}{2}\\[br]\ & = \frac{2\sin^2\alpha-2\sin^2 \beta}{2\sin \alpha \cos \alpha + 2\sin \beta \cos \beta } \\[br]\ & = \frac{1-\cos 2\alpha - (1-\cos 2\beta)}{\sin 2\alpha + \sin 2\beta}\\[br]\ & = \frac{1-\cos 2\alpha -1+\cos 2\beta)}{\sin 2\alpha + \sin 2\beta}\\[br]\ & = \frac{ \cos 2\beta-\cos 2\alpha}{\sin 2\alpha + \sin 2\beta}\\[br]\ & = \frac{2\sin \frac{2\alpha +2\beta}{2} \sin \frac{2\alpha -2\beta}{2} }{2\sin \frac{2\alpha + 2\beta}{2} \cos \frac{2\alpha -2\beta}{2}}\\[br]\ & = \frac{\sin (\alpha -\beta)}{\cos (\alpha -\beta) }\\[br]\ & = \tan (\alpha -\beta)\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]15. (a) Prove that: [math] \cos^3 x \sin^2 x =\frac{1}{16} ( 2\cos x-\cos3x - \cos 5x) [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos^3 x \sin^2 x \\[br]\ & = \cos^3 x \sin^2 x \\[br]\ & = \frac{3\cos x + \cos 3x}{4}\times \frac{1-\cos 2x}{2}\\[br]\ & = \frac{3\cos x - 3\cos 2x \cos x + \cos 3x - \cos 3x \cos 2x}{8}\times \frac{2}{2}\\[br]\ & = \frac{6\cos x - 3(2\cos 2x \cos x) + 2\cos 3x -2\cos 3x \cos 2x}{16}\\[br]\ & = \frac{6\cos x -3[\cos (2x+x) + \cos (2x-x)] +2\cos 3x -[\cos (3x+2x) +\cos(3x-2)]}{16}\\[br]\ & = \frac{6\cos x - 3\cos 3x - 3\cos x + 2\cos 3x - \cos 5x -\cos x }{16}\\[br]\ & = \frac{6\cos x - 3\cos x -\cos x - 3\cos 3x + 2\cos 3x - \cos 5x }{16}\\[br]\ & = \frac{1}{16}(2\cos x - \cos 3x -\cos 5x)\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]15. (b) Prove that: [math] \cos^4 \theta \sin^2 \theta = \frac{1}{32} ( 2+\cos 2\theta -\cos 6\theta -2\cos 4\theta) [/math][br][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos^4 \theta \sin^2 \theta \\[br]\ & = \cos^2 \theta \cos^2 \theta \sin^2 \theta\\[br]\ & = \cos^2 \theta\times \frac{1}{4} (2\sin \theta \cos \theta )^2\\[br]\ & = \frac{1}{4} \times \frac{1+\cos 2\theta}{2}\times (\sin 2\theta)^2\\[br]\ & = \frac{1+\cos 2\theta}{8} \times \sin^2 2\theta \\[br]\ & = \frac{1+\cos 2\theta}{8} \times \frac{1-\cos 4\theta}{2}\\[br]\ & = \frac{ 1-\cos 4\theta + \cos 2\theta - \cos 4\theta \cos 2\theta }{16}\\[br]\ & = \frac{ 1-\cos 4\theta + \cos 2\theta - \cos 4\theta \cos 2\theta }{16}\times \frac{2}{2} \\[br]\ & = \frac{ 2-2\cos 4\theta + 2\cos 2\theta - 2\cos 4\theta \cos 2\theta }{32}\\[br]\ & = \frac{ 2-2\cos 4\theta + 2\cos 2\theta - [\cos(4\theta +2\theta) + \cos (4\theta -2\theta)] }{32}\\[br]\ & = \frac{ 2-2\cos 4\theta + 2\cos 2\theta - \cos 6\theta - \cos 2\theta }{32}\\[br]\ & = \frac{1}{32} ( 2+ \cos 2\theta - \cos 6\theta -2\cos 4\theta )\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br][br]Alternative,[br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = \cos^4 \theta \sin^2 \theta \\[br]\ & = \left( \cos^2 \theta \right)^2 \sin^2 \theta \\[br]\ & = \left(\frac{1+ \cos2 \theta }{2} \right)^2 \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{ 1 + 2\cos 2\theta + \cos^2 2\theta}{4} \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{ 1 + 2\cos 2\theta + \cos^2 2\theta}{4}\times \frac{2}{2} \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{ 2 + 4\cos 2\theta + 2\cos^2 2\theta}{8} \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{ 2 + 4\cos 2\theta +(1+ \cos4\theta)}{8} \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{ 3 + 4\cos 2\theta + \cos4\theta}{8} \times \frac{1-\cos 2\theta}{2}\\[br]\ & = \frac{3 -3\cos 2\theta +4\cos 2\theta -4\cos^2 2\theta + \cos4\theta -\cos 4\theta \cos 2\theta}{16}\\[br]\ & = \frac{ 3+ \cos 2\theta -4\cos^2 2\theta +\cos 4\theta -\cos 4\theta \cos 2\theta}{16}\times \frac{2}{2}\\[br]\ & = \frac{ 6+ 2\cos 2\theta -4(2\cos^2 2\theta) +2\cos 4\theta -2\cos 4\theta \cos 2\theta}{32}\\[br]\ & = \frac{ 6 + 2\cos 2\theta -4( 1+\cos 4\theta ) + 2\cos 4\theta - [\cos (4\theta +2\theta) + \cos (4\theta -2\theta)]}{32}\\[br]\ & = \frac{ 6 + 2\cos 2\theta -4- 4\cos 4\theta + 2\cos 4\theta - [\cos 6\theta + \cos 2\theta ]}{32}\\[br]\ & = \frac{ 2 + 2\cos 2\theta - 2\cos 4\theta - \cos 6\theta - \cos 2\theta }{32}\\[br]\ & = \frac{ 2 + \cos 2\theta - \cos 6\theta - 2\cos 4\theta }{32}\\[br]\ & = \frac{1}{32}( 2 + \cos 2\theta - \cos 6\theta - 2\cos 4\theta )\\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br][br]15. (c) Prove that: [math] 16\cos^3 y =16\cos^5 y +2\cos y - \cos 3y -\cos 5y [/math][br]Solution:[br][math] \begin{align}[br]\text{RHS} \ & = 16\cos^5 y +2\cos y - \cos 3y -\cos 5y \\[br] \ & = 16\cos^5 y +2\cos y - (\cos 5y +\cos 3y ) \\[br] \ & = 16\cos^5 y +2\cos y - 2\cos \frac{5y + 3y}{2}\cos \frac{5y - 3y}{2} \\[br]\ & = 16\cos^5 y +2\cos y - 2\cos \frac{8y}{2}\cos \frac{2y}{2} \\[br]\ & = 16\cos^5 y +2\cos y - 2\cos 4y \cos y \\[br]\ & = 2\cos y [ 8\cos^4 y +1 -\cos 4y ]\\[br]\ & = 2\cos y [ 8 \cos^4 y +2\sin^2 2y]\\[br]\ & = 2\cos y [ 8 \cos^4 y + 2( \sin 2y )^2]\\[br]\ & = 2\cos y [ 8\cos^4 y +2 (2\sin y \cos y )^2 ]\\[br]\ & = 2\cos y [ 8\cos^4y +2\times 4\sin^2 y \cos^2 y]\\[br]\ & = 2\cos y [ 8\cos^4y +8\sin^2 y \cos^2 y ]\\[br]\ & = 2\cos y \times 8 \cos^2 y [\cos^2 y + \sin^2 y ]\\[br]\ & = 16 \cos^3 y [1]\\[br]\ & = 16 \cos^3 \\[br]\ & = \text{LHS}\\[br]\end{align} [/math][br][br]15. (d) Prove that: [math] 2\cos \frac{\pi^c}{13}\cos \frac{9\pi^c}{13}+\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13}=0 [/math][br]Solution:[br][math] \begin{align}[br]\text{LHS} \ & = 2\cos \frac{\pi^c}{13}\cos \frac{9\pi^c}{13}+\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br] \ & =\cos \left(\frac{\pi^c}{13} + \frac{9\pi^c}{13} \right)+\cos \left(\frac{\pi^c}{13} - \frac{9\pi^c}{13} \right)+\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & =\cos \left(\frac{10\pi^c}{13} \right)+\cos \left(\frac{-8\pi^c}{13} \right)+\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & =\cos \frac{10\pi^c}{13} +\cos \frac{8\pi^c}{13} +\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & =\cos \frac{13\pi^c-3\pi^c}{13} +\cos \frac{13\pi^c-5\pi^c}{13} +\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & = \cos \left (\pi^c - \frac{3\pi^c}{13} \right)+ \cos \left (\pi^c - \frac{5\pi^c}{13} \right) +\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & = -\cos \frac{3\pi^c}{13} - \cos \frac{5\pi^c}{13} +\cos \frac{3\pi^c}{13}+\cos \frac{5\pi^c}{13} \\[br]\ & = 0 \\[br]\ & = \text{RHS}\\[br]\end{align} [/math][br][br]16. (a) [math] \text{If } \sin (\alpha + \beta) = k\sin (\alpha -\beta), \text{ Prove that:} (k-1) \tan \alpha = (k+1) \tan \beta [/math][br]Solution:[br][br]Given,[br][math] \begin{align}[br]\ & \sin (\alpha + \beta) = k\sin (\alpha -\beta) \\[br]\ & \text{ or, }\frac{\sin (\alpha + \beta)}{\sin (\alpha -\beta) }= k\\[br]\ & \therefore k = \frac{\sin (\alpha + \beta)}{\sin (\alpha -\beta) }\\[br]\end{align} [/math][br]Now,[br][math] \begin{align}[br]\frac{k-1}{k+1} \ & = \frac{ \frac{\sin (\alpha + \beta)}{\sin (\alpha -\beta) }-1}{ \frac{\sin (\alpha + \beta)}{\sin (\alpha -\beta) }+1}\\[br]\ & = \frac{ \frac{\sin (\alpha + \beta)-\sin (\alpha -\beta)}{\sin (\alpha -\beta) }}{ \frac{\sin (\alpha + \beta)+\sin (\alpha -\beta)}{\sin (\alpha -\beta) }}\\[br]\ & = \frac{\sin (\alpha + \beta) - \sin (\alpha -\beta)}{\sin (\alpha +\beta) + \sin (\alpha -\beta)}\\[br]\ & = \frac{2\cos \alpha \sin \beta}{2\sin \alpha \cos \beta}\\[br]\ & = \frac{\cos \alpha }{\sin \alpha} \times \frac{\sin \beta}{\cos \beta}\\[br]\ & = \cot \alpha \tan \beta\\[br]\ & = \frac{1}{\tan \alpha} \times \tan \beta\\[br]\ & = \frac{\tan \beta }{\tan \alpha}\\[br]\text{ or, } \frac{k-1}{k+1} \ & = \frac{\tan \beta }{\tan \alpha}\\[br]\therefore (k-1) \tan \alpha \ & = (k+1) \tan \beta \text{ Proved}.\\[br]\end{align} [/math][br][br]16. (b)[math]\text{ If } \cos \theta = \cos \alpha . \cos \beta, \text{Prove that } \tan \frac{ \theta + \alpha} {2} . \tan \frac{\theta - \alpha}{2} = \tan^2 \frac{\beta}{2} [/math] [br]Solution:[br][math] \begin{align}[br] \text{Given,}\ & \\[br] \cos \theta \ & = \cos \alpha . \cos \beta\\[br]\text{or, } \frac{ \cos \theta }{\cos \alpha } \ & = \cos \beta \\[br]\text{or, } \frac{ \cos \theta }{\cos \alpha } \ & = \frac{\cos \beta}{1} \\[br]\text{or, } \frac{ \cos \alpha - \cos \theta }{ \cos \alpha + \cos \theta} \ & = \frac{ 1-\cos \beta}{1+\cos \beta} \\[br]\text{or, }\frac{2\sin \frac{\alpha + \theta}{2} \sin \frac{\theta - \alpha}{2}} {2\cos \frac{\alpha + \theta}{2} \cos \frac{\alpha - \theta}{2} } \ & = \frac{1-(1-2\sin^2 \frac{\beta}{2}) }{1+ (2cos^2\frac{\beta}{2} -1)}\\[br]\text{or, }\frac{\sin \frac{\alpha + \theta}{2}} {\cos \frac{\alpha + \theta}{2} } \times \frac{\sin \frac{\theta -\alpha}{2}}{\cos \frac{\theta - \alpha}{2}} \ & = \frac{2\sin^2 \frac{\beta}{2} }{2cos^2\frac{\beta}{2} }\\[br] \text{ or, } \tan \frac{\alpha + \theta}{2}. \tan \frac{\theta - \alpha}{2} \ & = \frac{\sin^2 \frac{\beta}{2}}{\cos^2\frac{\beta}{2}}\\[br]\therefore \tan \frac{\alpha + \theta}{2}. \tan \frac{\theta - \alpha}{2} \ & = \tan^2 \frac{\beta}{2}\\[br]\end{align} [/math][br][br]