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General Solutions of (sin theta)^2 = (sin alpha)^2 , (cos theta)^2 = (cos alpha)^2 , (tan theta)^2 = (tan alpha)^2

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Question

Find the general solution:
$s i n x + s i n 3 x + s i n 5 x = 0$

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Solution

$s i n x + s i n 3 x + s i n 5 x = 0$
$\Rightarrow s i n x + s i n 5 x + s i n 3 x = 0$
$\Rightarrow 2 sin (\frac{x + 5 x}{2}) cos (\frac{x - 5 x}{2}) + s i n 3 x = 0 \Rightarrow sin 3 x (2 cos 2 x + 1) = 0$
$\Rightarrow sin 3 x = 0$ or $cos 2 x = \frac{- 1}{2}$
$\Rightarrow 3 x = n π$ or $2 x = 2 n π \pm \frac{π}{3}$
$\Rightarrow x = \frac{n π}{3}$ , $n π \pm \frac{π}{6}$

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sin x + sin 3 x + sin 5 x = 0

sin x + sin 3 x + sin 5 x = 0

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sin x + sin 3x + sin 5x = 0

sin x + sin 3 x + sin 5 x = 0

s i n x + s i n 5 x = s i n 2 x + s i n 4 x

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