Mathematics

Problem

(i) Obtain expressions for

\sin 4x

and

\cos 4x

entirely in terms of

\sin x

and

\cos x

. (ii) Hence prove that

\sin^{4}x=\dfrac18\bigl(3-4\cos 2x+\cos 4x\bigr).

(iii) Using the identity proved in part (ii), find the exact value of

\sin^{4}15^{\circ}