Question

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

$\sin \frac{3\vartheta }{2}\cos \frac{\vartheta }{2}$

Short answer

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

Step-by-step solution

Step 1. Given expression 

$\sin \frac{3\vartheta }{2}\cos \frac{\vartheta }{2}$

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

Using identity :

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

We get :

role="math" localid="1646448141774" $$\begin{gathered} \sin \frac{3\vartheta }{2}\cos \frac{\vartheta }{2}=\frac{1}{2}\left[\sin \right(\frac{3\vartheta }{2}+\frac{\vartheta }{2})+\sin (\frac{3\vartheta }{2}-\frac{\vartheta }{2}) \\ \sin \frac{3\vartheta }{2}\cos \frac{\vartheta }{2}=\frac{1}{2}[\sin \left(2\vartheta \right)+\sin \left(\vartheta \right)] \end{gathered}$$