Mathematics

Problem

For any real angles

a

and

b

, define

S=\sin a + \sin b

. a. (i) Prove that

S = 2\sin\dfrac{a+b}{2}\cos\dfrac{a-b}{2}

. b. (ii) By following analogous steps, obtain an expression for

\sin a - \sin b

in terms of half-sum and half-difference of

a

and

b

. c. (iii) Hence simplify the numerical expression

\sin 65^{\circ}+\sin 25^{\circ}