Solve :
$sin m x + sin n x = 0$

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Solution

$s i n (m x) + s i n (n x) = 0$
$2 s i n \frac{(m + n)}{2} x . c o s \frac{(m - n) x}{2} = 0$
$∵ s i n A + s i n B = 2 s i n (\frac{A + B}{2}) . c o s (\frac{A - B}{2})$
$∴ s i n [\frac{(m + n) x}{2}] = 0$ or $c o s [\frac{(m - n) x}{2}] = 0$
$∴ \frac{(m + n) x}{2} = p x$ or $\frac{(m - n) x}{2} = (2 q + 1) \frac{x}{2}$
$p . q \in I$
$x = \frac{2 p x}{m + n}$ or $x = \frac{(2 q + 1) x}{m - n}$

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