Mathematics

Problem

Answer the following questions related to the sum of fourth powers of sine and cosine. a. (i) Prove that

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

. b. (ii) Using the result obtained in part (i), find the exact value of

\sin^{4}60^{\circ} + \cos^{4}60^{\circ}