Mathematics - Trigonometric Equation Question with Solution | TestHub

$\begin{array}{rl}& \text{ Given, }\sin 9\theta =\sin \theta \\ & \Rightarrow \sin 9\theta -\sin \theta =0\\ & \Rightarrow 2\cos \left(\frac{9\theta +\theta }{2}\right)\sin \left(\frac{9\theta -\theta }{2}\right)=0\\ & \left[\because \sin C-\sin D=2\cos \left(\frac{C+D}{2}\right)\sin \left(\frac{C-D}{2}\right)\right]\\ & \Rightarrow 2\cos 5\theta \sin 4\theta =0\\ & \Rightarrow \text{ Either }\cos 5\theta =0\text{ or }\sin 4\theta =0\text{, also }\theta \in [0,2\pi ]\\ & \Rightarrow 5\theta =(2n+1)\frac{\pi }{2}\\ & \text{ or }4\theta =n\pi \\ & \Rightarrow \theta =(2n+1)\frac{\pi }{10}\text{ or }\theta =\frac{n\pi }{4}\\ & \therefore \theta =\frac{\pi }{10},\frac{3\pi }{10},\frac{\pi }{2},\frac{7\pi }{10},\frac{9\pi }{10},\frac{11\pi }{10},\frac{13\pi }{10},\frac{3\pi }{2},\frac{17\pi }{10},\frac{19\pi }{10}\\ & \text{ or }0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4},2\pi \\ & \therefore \text{ Total number of solutions }=17\\ & \left(\because \frac{\pi }{2}\text{ and }\frac{3\pi }{2}\text{ are common }\right)\end{array}$