Answer
See below:
Work Step by Step
(b)
In order to verify the equations, use the trigonometric identity.
$\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta $
Now, apply the above identity in the following equation:
$\cos \left( x+\frac{\pi }{2} \right)=\cos x\cos \frac{\pi }{2}-\sin x\sin \frac{\pi }{2}$
By using the values $\cos \frac{\pi }{2}=0$ and $\sin \frac{\pi }{2}=1$, we get:
$\begin{align}
& \cos \left( x+\frac{\pi }{2} \right)=\cos x\cos \frac{\pi }{2}-\sin x\sin \frac{\pi }{2} \\
& =\cos x.\left( 0 \right)-\sin x.\left( 1 \right) \\
& =0-\sin x \\
& =-\sin x
\end{align}$
Hence, the equation $\cos \left( x+\frac{\pi }{2} \right)$ is equal to $-\sin x$.
