Question

How do I solve this problem? Prove Identities or simplify using sum and difference formula.

Answer 1

One of the trigonometric identities is :

$\sin (A+B)=\sin A\cos B+\cos A\sin B$

From the problem, we have :

$\sin (\frac{3\pi}{2}+x)$

Applying the identity above,

$\sin (\frac{3\pi}{2}+x)=\sin \frac{3\pi}{2}\cos x+\cos \frac{3\pi}{2}\sin x$

Note that :

$\begin{gathered} \sin \frac{3\pi}{2}=-1 \\ \text{and} \\ \cos \frac{3\pi}{2}=0 \end{gathered}$

The identity will be :

$\begin{gathered} \sin \frac{3\pi}{2}\cos x+\cos \frac{3\pi}{2}\sin x \\ \Rightarrow(-1)(\cos x)+(0)(\sin x) \\ \Rightarrow-\cos x \end{gathered}$

The answer is -cos x