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Chapter 5: Problem 97

Verify that \(\cos ^{2} x-\sin ^{2} x=\cos 2 x\) by using a product-to-sum identity.

Short Answer

Expert verified
The original identity \(\cos ^{2} x - \sin ^{2} x = \cos 2x\) has been verified by using the product-to-sum identities.

Step by step solution

01

Recognize the Product-to-Sum Identity

The proper product-to-sum formula that will be used is \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\) and \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).
02

Substitute into Identity

Substitute the product-to-sum identities from Step 1 into the original equation to get \(\cos^2(x) - \sin^2(x) = \frac{1 + \cos(2x)}{2} - \frac{1 - \cos(2x)}{2}\).
03

Simplify

Simplifying the result of step 2 yields: \(\cos^2(x) - \sin^2(x) = \cos(2x)\). This verifies the original identity.

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