Mathematics

Problem

Let

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

. Answer the questions that follow. a. (i) Find the exact values of

\sin 75^{\circ}

\cos 15^{\circ}

\cos 75^{\circ}

and

\sin 15^{\circ}

, expressing each in simplest radical form. b. (ii) Using the values obtained in part (i), evaluate

S

exactly. c. (iii) Show that

S=1

and explain, with reference to a general trigonometric identity, why this result is expected. d. (iv) Using

S

, simplify the expression

T=\frac{\sin 75^{\circ}}{\cos 15^{\circ}}+\frac{\cos 75^{\circ}}{\sin 15^{\circ}}

so that

T

is expressed as a rational number.