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Chapter 5: Problem 30

Use a double-angle or half-angle identity to verify the given identity. $$\sin ^{2} x+\cos 2 x=\cos ^{2} x$$

Short Answer

Expert verified
The given identity \(\sin ^{2} x+\cos 2 x=\cos ^{2} x\) is correct as it has been verified by using the double angle identity of cosine and the Pythagorean identity.

Step by step solution

01

Substitute the double-angle identity for cosine

Start by using the identity \(\cos 2x = 1- 2\sin^2 x\) and substitute \(\cos 2x\) in the given equation. This results in the equation \(\sin ^{2} x+ 1- 2\sin^2 x = \cos ^{2} x\). Thus, the problem transforms into verifying this equality.
02

Simplify left side of the equation

Simplify the left side of the equation by combining like terms. This gives \(1 - \sin^{2}x = \cos^2x\) which is a simplified equation to verify.
03

Utilize Pythagorean Identity

The Pythagorean identity states that \( \sin^2x + \cos^2x = 1\). This means that \(\cos^2x\) can be replaced with \(1 - \sin^2x\). Substituting this back into the equation gives \(1 - \sin^{2}x = 1 - \sin^{2}x\), thus verifying the identity.

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