Mathematics

The key is to recognize that $\sin^4 x$ is a high power that's hard to work with directly—but you can reduce it by repeatedly applying the power-reducing identity $\sin^2 x = \frac{1 - \cos 2x}{2}$. Square this identity once to get $\sin^4 x$ in terms of $\cos 2x$, then apply the power-reducing identity again to eliminate $\cos^2 2x$ and express everything in terms of $\cos 4x$. This transforms a fourth power into a clean linear combination of cosines.

The key is to recognize that $\sin^4 x$ is a high power that's hard to work with directly—but you can reduce it by repeatedly applying the power-reducing identity $\sin^2 x = \frac{1 - \cos 2x}{2}$. Square this identity once to get $\sin^4 x$ in terms of $\cos 2x$, then apply the power-reducing identity again to eliminate $\cos^2 2x$ and express everything in terms of $\cos 4x$. This transforms a fourth power into a clean linear combination of cosines.