Question

Number of solutions of the equation $\sin 5\theta \cos 3\theta =\sin 9\theta \cos 7\theta$ in $\left[\begin{array}{cc}0,& \frac{\pi }{4}\end{array}\right]$ is .................

Answer 1

$\sin 5\theta \cos 3\theta =\sin 9\theta \cos 7\theta \Rightarrow 2\sin 5\theta \cos 3\theta =2\sin 9\theta \cos 7\theta \Rightarrow \sin 8\theta +\sin 2\theta =\sin \left(16\theta \right)+\sin 2\theta \Rightarrow \sin 8\theta -\sin 16\theta =0\Rightarrow \sin 8\theta (1-2\cos 8\theta )=0$
$\Rightarrow \sin 8\theta =0$ or $\cos 8\theta =\frac{1}{2}$
$\Rightarrow 8\theta =n\pi$ or $8\theta =2n\pi \pm \frac{\pi }{3}$
$\Rightarrow \theta =\frac{n\pi }{8}$ or $\theta =\frac{2n\pi }{8}\pm \frac{\pi }{24}$
Therefore number of values of $\theta$ is 5