Answer
The required solution is True.
Work Step by Step
Let us consider the left side of the given expression:
$\cos \frac{\pi }{2}\cos \frac{\pi }{3}$
Now, put the values of the trigonometric functions to find the exact value as shown below:
$\begin{align}
& \cos \frac{\pi }{2}\cos \frac{\pi }{3}=0\cdot \frac{1}{2} \\
& =0
\end{align}$
Also, consider the right side of the given expression:
$\frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)+\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]$
Then, put the values of the trigonometric functions to find the exact value as shown below:
$\begin{align}
& \frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)+\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]=\frac{1}{2}\left[ \cos \left( \frac{3\pi -2\pi }{6} \right)+\cos \left( \frac{3\pi +2\pi }{6} \right) \right] \\
& =\frac{1}{2}\left[ \cos \left( \frac{\pi }{6} \right)+\cos \left( \frac{5\pi }{6} \right) \right] \\
& =\frac{1}{2}\cdot \left[ \frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2} \right] \\
& =0
\end{align}$
Thus, the left side of the expression is equal to the right side, which is $\cos \frac{\pi }{2}\cos \frac{\pi }{3}=\frac{1}{2}\left[ \cos \left( \frac{\pi }{2}-\frac{\pi }{3} \right)-\cos \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right]$.
