Mathematics

Problem

Establish the following trigonometric identity for any real

\theta

. a. (i) Express

\sin^{4}\theta + \cos^{4}\theta

in terms of

\sin^{2}\theta

and

\cos^{2}\theta

. b. (ii) Using the result of part (i) and a suitable double–angle relation, prove that

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