Answer
See the full explanation below.
Work Step by Step
Let us consider the left side of the given expression:
$\sin \left( x+\frac{3\pi }{2} \right)$
By using the identity of trigonometry, $\sin \,\left( \alpha +\beta \right)=\sin \,\alpha \,\cos \beta +\cos \,\alpha \,\sin \,\beta $, the above expression can be further simplified as:
$\begin{align}
& \sin \left( x+\frac{3\pi }{2} \right)=\sin \,x\,\cos \,\frac{3\pi }{2}+\cos \,x\,\sin \,\frac{3\pi }{2} \\
& =\sin \,x\times 0+\cos \,x\times -1 \\
& =0-\cos \,x \\
& =-\cos \,x
\end{align}$
Thus, the left side of the expression is equal to the right side, which is $\sin \left( x+\frac{3\pi }{2} \right)=-\cos \,x$.
