Question

Express the product $\sin \left(4\theta \right)\sin \left(2\theta \right)$as a sum containing only sines or only cosines.

Short answer

$\sin \left(4\theta \right)\sin \left(2\theta \right)=\frac{\cos \left(2\theta \right)-\cos \left(6\theta \right)}{2}$

Step-by-step solution

Step 1. Given expression

$\sin \left(4\theta \right)\sin \left(2\theta \right)$

Step 2. Rewriting the expression   

$$\begin{gathered} 2\sin A\sin B=\cos (A-B)-\cos (A+B) \\ So, \\ \sin \left(4\theta \right)\sin \left(2\theta \right)=\frac{\cos \left(2\theta \right)-\cos \left(6\theta \right)}{2} \end{gathered}$$