Simplify the trigonometric expression cos (2x) + 1 using double-angle identities.

A. - 2sin^2 (x)

B. 2cos^2 (x) - 2sin^2 (x)

C. 2cos^2 (x) + 2sin^2 (x)

D. 2cos^2 (x)

The answer is 2cos^2 (x)

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Since the cosine double identity can be expressed as

cos (2x) = cos²x - sin²x

we have

cos (2x) + 1 = cos²x - sin²x + 1

We can rearrange it so it becomes

= cos²x + 1 - sin²x

Note that 1 - sin²x = cos²x. This is because if we take the Pythagorean identity, sin²x + cos²x = 1, and subtract both sides by sin²x, we end up with cos²x = 1 - sin²x.

Therefore

= cos²x + (1 - sin²x)

= cos²x + cos²x

= 2cos² (x)