Find the general solution of the equation $sin 6 x = sin 4 x - sin 2 x$

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Solution

We have, $sin 6 x = sin 4 x - sin 2 x$

$\Rightarrow 2 sin 3 x cos 3 x = 2 cos (\frac{4 x + 2 x}{2}) sin (\frac{4 x - 2 x}{2})$

$\Rightarrow 2 sin 3 x cos 3 x = 2 cos 3 x sin x$

$\Rightarrow 2 cos 3 x (sin 3 x - sin x) = 0$

$\Rightarrow cos 3 x = 0, sin 3 x = sin x$

When $cos 3 x = 0$

$\Rightarrow 3 x = (2 n + 1) \frac{π}{2}$

$\Rightarrow x = (2 n + 1) \frac{π}{6}$

When $sin 3 x = sin x$

$∴ 3 x = n π + {(- 1)}^{n} x$

When $n$ is even, $x = n π$

When $n$ is odd, $x = (2 n + 1) \frac{π}{4}$

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x

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