Mathematics

Problem

Establish an identity connecting the fourth powers of sine and cosine with a double–angle term. a. (i) Starting with

\sin^4 \theta + \cos^4 \theta,

express the sum in terms of

\sin^2 \theta

and

\cos^2 \theta

only. b. (ii) Further rewrite your result completely in terms of

\sin 2\theta

and hence prove that

\sin^4 \theta + \cos^4 \theta = 1 - \dfrac{1}{2}\sin^2 2\theta.