Question

Prove that

(i) 2sin5π/12 sinπ/12 = 1/2

(ii) 2cos5π/12 cosπ/12 = 1/2

(iii) 2sin5π/12 cosπ/12 = (2 + √3)/2

Answer 1

(i) 2sin\(\frac{5\pi}{12}.\) sin\(\frac{\pi}{12}\)

………[Using –2sinx.siny = cos(x + y)–cos (x–y)]

………..[using 2cosx.cosy = cos(x + y) + cos(x–y)]

= \(\frac{1}{2}\)

...[Using2sinx.cosy = sin(x + y) + sin(x–y)]

= 1 + \(\frac{\sqrt{3}}{2}\Rightarrow \frac{2\,+\,\sqrt{3}}{2}\)