Solve the following equation:
$sin 5 x + sin x + 2 {sin}^{2} x = 1$

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Solution

$sin 5 x + sin x = 1 - {2 sin}^{2} x$
$2 sin 3 x cos 2 x = cos 2 x$
Either, $cos 2 x = 0, 2 x = (2 n + 1) \frac{π}{2}, x = (2 n + 1) \frac{π}{4} .$
or, $sin 3 x = \frac{1}{2}, 3 x = n π + {(- 1)}^{n} \frac{π}{6}, x = n \frac{π}{3} + {(- 1)}^{n} \frac{π}{18} .$

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