Complete the identity cos4 x sin4 x A 1 2cos2x B 1 2sin2x

Complete the identity. cos^4 x - sin^4 x = ? A. 1 -2cos^2x B. 1 -2sin^2x C. 1 + 2sin^2x D. 1 + 2cos^2X Complete the identity.

cos^4 x - sin^4 x

we can write it as

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

applying difference of squares formula

a^2 - b^2 = (a-b)(a+b)

(cos^2x)^2 - (sin^2x )^2 = (cos^2x + sin^2x )( cos^2 x - sin^2x)

cos^2 x + sin^2 x = 1

cos^2x - sin^2 x = (1- sin^2 x ) - sin^2 x = 1 - 2 sin^2 x

plugging the values

(cos^2x + sin^2x )( cos^2 x - sin^2x) = 1(1-2sin^2x )

= 1- 2 sin^2x

option b is correct