Answer
The required solution is True.
Work Step by Step
Let us consider the left side of the given expression:
$sin\pi \cos \frac{\pi }{2}$
Then, put the values of the trigonometric functions to find the exact value as shown below:
$\begin{align}
& sin\pi \cos \frac{\pi }{2}=0\cdot 0 \\
& =0
\end{align}$
Also, consider the right side of the given expression:
$\frac{1}{2}\left[ \sin \left( \pi +\frac{\pi }{2} \right)+\sin \left( \pi -\frac{\pi }{2} \right) \right]$
Now, put the values of the trigonometric functions to find the exact value as shown below:
$\begin{align}
& \frac{1}{2}\left[ \sin \left( \pi +\frac{\pi }{2} \right)+\sin \left( \pi -\frac{\pi }{2} \right) \right]=\frac{1}{2}\left[ \sin \left( \frac{2\pi +\pi }{2} \right)+\sin \left( \frac{2\pi -\pi }{2} \right) \right] \\
& =\frac{1}{2}\left[ \sin \left( \frac{3\pi }{2} \right)+\sin \left( \frac{\pi }{2} \right) \right] \\
& =\frac{1}{2}\left[ -1+1 \right] \\
& =0
\end{align}$
So, the left side of the expression is equal to the right side, which is $sin\pi \cos \frac{\pi }{2}=\frac{1}{2}\left[ \sin \left( \pi +\frac{\pi }{2} \right)+\sin \left( \pi -\frac{\pi }{2} \right) \right]$.
