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Trigonometric Ratios of Compound Angles

Solve the fol...

Question

Solve the following equation:
$cos (\frac{π}{2} + 5 x) + sin x = 2 cos 3 x$

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Solution

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

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