Question

Express the product $\sin 4\theta \cos 6\theta$as a sum containing only sines or only cosines.

Short answer

$\sin 4\theta \cos 6\theta =\frac{\sin \left(10\theta \right)-\sin \left(2\theta \right)}{2}$

Step-by-step solution

Step 1. Given expression   

$\sin 4\theta \cos 6\theta$

Step 2. Rewriting the expression      

$$\begin{gathered} 2\sin A\cos B=\sin (A+B)+\sin (A-B) \\ so, \\ \sin 4\theta \cos 6\theta =\frac{\sin \left(10\theta \right)+\sin (-2\theta )}{2} \\ =\frac{\sin \left(10\theta \right)-\sin \left(2\theta \right)}{2}\left(\sin \right(-\theta )=-\sin \theta \end{gathered}$$