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Q.
The general solution of equation $\sin x+\sin 5 x=\sin 2 x+\sin 4 x$ is :
Trigonometric Functions
Solution:
$\sin x+\sin 5 x=\sin 2 x+\sin 4 x $
$\Rightarrow 2 \sin 3 x \cos 2 x=2 \sin 3 x \cos x$
$\Rightarrow \sin 3 x=0$
or $\cos 2 x=\cos x $
$\Rightarrow 3 x=n \pi$
or $2 x=2 n \pi \pm x$
$\Rightarrow x=\frac{n \pi}{3}, 2 n \pi, \frac{2 n \pi}{3}$
$\Rightarrow x=\frac{n \pi}{3}$ (It includes all three possible solutions)