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)===since the cosine double identity can be expressed as    cos(2x) = cos²x - sin²xwe have    cos(2x) +1 = cos²x - sin²x + 1we can rearrange it so it becomes    = cos²x + 1 - sin²xnote 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)