Question

If $\tan 3\theta .\tan 2\theta =1$ then find the value of $\sin 5\theta$.

Answer 1

Given,

$\tan 3\theta .\tan 2\theta =1$

or, $\sin 3\theta .\sin 2\theta =\cos 3\theta .\cos \theta$

or, $\cos 3\theta .\cos 2\theta -\sin 3\theta .\sin 2\theta =0$

or, $\cos (3\theta +2\theta )=0$

or, $\cos 5\theta =0$

$\Rightarrow \sin 5\theta =1$ [ Since ${\sin }^{2}5\theta +{\cos }^{2}5\theta =1$]