Question

express each product as a sum containing only sines or only cosines

$\cos 3\vartheta \cos 4\vartheta$

Short answer

$\cos 3\vartheta \cos 4\vartheta =\frac{1}{2}\left[\cos \right(7\vartheta )+\cos (\vartheta \left)\right]$

Step-by-step solution

Step 1. Given expression 

$\cos 3\vartheta \cos 4\vartheta$

Step 2. expressing them in terms of single sine or cosine

Using identity :

$\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

We get :

$$\begin{gathered} \cos 3\vartheta \cos 4\vartheta =\frac{1}{2}\left[\cos \right(3\vartheta +4\vartheta )+\cos (4\vartheta -3\vartheta \left)\right] \\ \cos 3\vartheta \cos 4\vartheta =\frac{1}{2}\left[\cos \right(7\vartheta )+\cos (\vartheta \left)\right] \end{gathered}$$