<s>[INST] Find the number of solutions of the equation $\sin^22x-\cos^28x=\frac{1}{2}\cos10x$ Find the number of solutions of the equation $\sin^22x-\cos^28x=\frac{1}{2}\cos10x$ lying in the interval $(0,\frac{\pi}{2})$ 

I found the period of $\sin^22x-\cos^28x$ as $\pi$ and the period of $\frac{1}{2}\cos10x$ is $\frac{\pi}{5}$
I do not know how to solve it further.
 [/INST] $$\sin^2 2x-\cos^2 8x=\frac{1}{2} \cos10x$$
Using the formulae, $$\sin^2\alpha=\frac{1-\cos 2 \alpha}{2}; \cos^2\alpha=\frac{1+\cos 2 \alpha}{2}$$
$$\frac{1-\cos 4x}{2}-\frac{1+\cos 16x}{2}=\frac{1}{2} \cos10x$$
$$-\cos 4x-\cos 16 x=\cos 10 x$$
$$2 \cos 10x \cos 6x + \cos 10x=0$$
$\cos 10x =0$ or $\cos 6x= -\frac 12$
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