Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$(\sin x+\sin 4x)+(\sin 2x+\sin 3x)=0$ $\Rightarrow 2\sin \frac{5x}{2}\cdot \cos \frac{3x}{2}+2\sin \frac{5x}{2}\cdot \cos \frac{x}{2}=0$ $\Rightarrow 2\sin \frac{5x}{2}\left\{\cos \frac{3x}{2}+\cos \frac{x}{2}\right\}=0$ $\Rightarrow 2\sin \frac{5x}{2}\left\{2\cos x\cos \frac{x}{2}\right\}=0$ $2\sin \frac{5x}{2}=0\Rightarrow \frac{5x}{2}=0,\pi ,2\pi ,3\pi ,4\pi ,5\pi ,\dots$ $\Rightarrow x=0,\frac{2\pi }{5},\frac{4\pi }{5},\frac{6\pi }{5},\frac{8\pi }{5},2\pi$ $\cos \frac{x}{2}=0\Rightarrow \frac{x}{2}=\frac{\pi }{2}\Rightarrow x=\pi ;$ $\cos x=0\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{2}$ So, sum $=6\pi +\pi +2\pi =9\pi$