# Express $f(\theta)=6\cos2\theta+5(\sin2\theta)^2+3$ as a sum of powers of $\sin\theta$

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

$$f(\theta)=6\cos2\theta+5(\sin2\theta)^2+3$$

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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

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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