How to simplfiy $ \cos^{2}(\theta) \sin(\theta) $ to terms of $\sin$ or $\cos$ only?

Question

Is it possible to simplify $$ \cos^{2}(\theta) \sin(\theta) $$ to terms of only $\sin$ or $\cos$?

I need to simplify this to take the integral.

Answer 1

If you intend to integrate it with respect to $\theta$, you definitely do not want to simplify it to get only sines or cosines. Make the substitution $u=\cos\theta$ instead. Then $du=-\sin\theta d\theta$, so $$\cos^2\theta\sin\theta d\theta=u^2(-du)=-u^2 du\;,$$ which is easy to integrate.

Answer 2

You can write $\cos^2(\theta)=1-\sin^2(\theta)$ and you have $\sin (\theta)-\sin^3 (\theta)$ but as Brian M. Scott says, that is a step backward for integration.