Question

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

$\sin 4\vartheta -\sin 2\vartheta$

Short answer

$2\cos \left(3\vartheta \right)\sin \left(\vartheta \right)$

Step-by-step solution

Step 1. Given Information

$\sin 4\vartheta -\sin 2\vartheta$

Step 2. Solving using identity

We know that :

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

Using it :

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