Question

The number of distinct solutions of $\sin 5\theta \cos 3\theta =\sin 9\theta \cos 7\theta in\left[0,\frac{\pi }{2}\right]$ is

• A. 4
• B. 5
• C. 8
• D. 9

Answer 1

Answer: (D)

$\sin 5\theta \cos 3\theta =\sin 9\theta \cos 7\theta$
$\Rightarrow 2\sin 5\theta \cos 3\theta =2\sin 9\theta \cos 7\theta$
$\Rightarrow \sin 8\theta +\sin 2\theta =\sin 16\theta +\sin 2\theta$
$\Rightarrow \sin 16\theta =\sin 8\theta \Rightarrow 16\theta =n\pi +(-1{)}^{n}8\theta$
$\Rightarrow$ 16$\theta =2m\pi +8\theta$
[If $n$ is even and $n=2m$]
$\Rightarrow 8\theta =2m\pi \Rightarrow \theta =\frac{m\pi }{4}$
and $16\theta =(2m+1)\pi -8\theta$
[If $n$ is odd and $n=2m+1$]
$\Rightarrow$ $24\theta =(2m+1)\pi \Rightarrow \theta =\frac{2m+1}{24}\pi$
Thus, $\theta =\frac{m\pi }{4},\frac{2m+1}{24}\pi$ where $m\in Z$
$=0,\frac{\pi }{4},\frac{\pi }{2},\frac{\pi }{24},\frac{7\pi }{24},\frac{3\pi }{24},\frac{11\pi }{24}$
$\therefore$ total no. of solutions = 9