Mathematics

Problem

Evaluate the following expressions exactly, showing each step clearly and without using a calculator. a. (i)

S=\sin 75^{\circ}\,\cos 15^{\circ}+\cos 75^{\circ}\,\sin 15^{\circ}

. b. (ii)

P=\sin 75^{\circ}\,\cos 15^{\circ}-\cos 75^{\circ}\,\sin 15^{\circ}

. c. (iii) Verify that the value obtained in part (ii) equals

\sin 60^{\circ}