The potential energy in joules of an oscillating spring can be represented by the expression [math]16k[\frac{\csc^2(2\pi t)-1}{\csc^2(2\pi t)}][/math], where [math]k[/math] is a constant and [math]t[/math] is the time in seconds since the spring began oscillating. Create a two-column proof to show that the potential energy of the spring can also be written as [math]16k \cos^2 (2\pi t)[/math].
[list=1] [*]Create the outline for the two-column proof. [*]Rewrite the fraction on the left side of the equation as the difference of two fractions. [*]Simplify [math]\frac{\csc^2(2\pi t)}{\csc^2(2\pi t)}[/math]. [*]Use the reciprocal identity [math]\sin\theta=\frac{1}{\csc\theta}[/math] to rewrite the expression from step 3. [*]Use the Pythagorean identity [math]\sin^2\theta + \cos^2\theta=1[/math] to rewrite the expression [math]16k[1-\sin^2(2\pi t)][/math]. [*]Fill in the columns of the two-column proof. [/list]