Question

Sinteta+sin3teta+sin5teta=sin^2 3 teta/sinteta

Answer 1

L.H.S.=
\[(\sin 5 \theta+\sin \theta)+\sin 3\theta=2\sin \frac{ 5\theta+\theta }{ 2 }\cos \frac{ 5\theta-\theta }{ 2}+\sin 3\theta\]
\[=2\sin 3\theta \cos 2 \theta+\sin 3\theta =\sin 3\theta \left( 1+2\cos 2\theta \right)\]
\[=\sin 3\theta \left\{ 1\left( \cos ^2\theta+\sin ^2\theta \right)+2\left( \cos ^2\theta-\sin ^2\theta \right) \right\} ...(1)\]
\[R.H.S.=\frac{ \sin ^23\theta }{\sin \theta }=\frac{ \sin 3\theta \sin 3\theta }{ \sin \theta }\]
\[=\frac{ \sin 3\theta \left( 3\sin \theta-4\sin ^3\theta \right) }{ \sin \theta }=\sin 3\theta \left( 3-4\sin ^2\theta \right)\]
\[=\sin 3\theta \left( 3\left( \cos ^2\theta+\sin ^2\theta \right) -4\sin ^2\theta \right)\]
\[=\sin 3\theta \left( 3\cos ^2\theta-\sin ^2\theta \right) ...(2)\
from (1) and (2)
L.H.S.=R.H.S