The potential energy in joules of an oscillating spring can be represented by the expression , where is a constant and 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 .

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 .

Use the reciprocal identity to rewrite the expression from step 3.

Use the Pythagorean identity to rewrite the expression .

Fill in the columns of the two-column proof.

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