Question

express each sum or difference as a product of sines and/or cosines

$\cos 2\vartheta +\cos 4\vartheta$

Short answer

$2\cos 3\vartheta \cos \vartheta$

Step-by-step solution

Step 1. Given information

$\cos 2\vartheta +\cos 4\vartheta$

Step 2. Solving using identity

We know that :

$\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$

Using it :

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