Mathematics

The key is to recognize that you're differentiating a composite function, so the chain rule will create terms involving $\cos(\sin x)$ that persist through both derivatives. Once you have $y'$ and $y''$, the trick is to factor out common expressions—particularly $\cos(\sin x)$—and use the identity $\sin^2(\sin x) + \cos^2(\sin x) = 1$ to simplify the resulting combination. This transforms what looks messy into an elegant identity.

The key is to recognize that you're differentiating a composite function, so the chain rule will create terms involving $\cos(\sin x)$ that persist through both derivatives. Once you have $y'$ and $y''$, the trick is to factor out common expressions—particularly $\cos(\sin x)$—and use the identity $\sin^2(\sin x) + \cos^2(\sin x) = 1$ to simplify the resulting combination. This transforms what looks messy into an elegant identity.