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Please explain how to express the below formula as a sum of powers of $\sin \theta$


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double angle formulas – vadim123 Jul 29 '13 at 4:02
What have you tried so far? Are you familiar with the double angle formulae? – Alex Wertheim Jul 29 '13 at 4:02
up vote 2 down vote accepted

Note that $$\cos(2\theta)=\cos^{2}(\theta)-\sin^{2}(\theta)=1-2\sin^{2}(\theta)$$ and $$\sin^{2}(2\theta)=4\sin^{2}(\theta)\cos^{2}(\theta)=4\sin^{2}(\theta)(1-\sin^{2}(\theta)).$$

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Hint: First convert from $\sin 2\theta$ to $\cos 2\theta$ by using the identity:

$$ \sin^2 2\theta+\cos^2 2\theta=1 $$

Then convert from $\cos 2\theta$ to $\sin \theta$ by using the identity:

$$ \cos 2\theta = 1-2\sin^2 \theta $$

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