Answer
The required solution is True.
Work Step by Step
Let us consider the left side of the given expression:
$\sin {{60}^{\circ }}\sin {{30}^{\circ }}$
Now, put the values of the trigonometric functions to find the exact value as given below:
$\begin{align}
& \sin {{60}^{\circ }}\sin {{30}^{\circ }}=\frac{\sqrt{3}}{2}\cdot \frac{1}{2} \\
& =\frac{\sqrt{3}}{4}
\end{align}$
Also, consider the right side of the given expression:
$\frac{1}{2}\left[ \cos \left( {{60}^{\circ }}-{{30}^{\circ }} \right)-\cos \left( {{60}^{\circ }}+{{30}^{\circ }} \right) \right]$
Then, put the values of the trigonometric functions to find the exact value as shown below:
$\begin{align}
& \frac{1}{2}\left[ \cos \left( {{60}^{\circ }}-{{30}^{\circ }} \right)-\cos \left( {{60}^{\circ }}+{{30}^{\circ }} \right) \right]=\frac{1}{2}\left[ \cos {{30}^{\circ }}-\cos {{90}^{\circ }} \right] \\
& =\frac{1}{2}\left[ \frac{\sqrt{3}}{2}-0 \right] \\
& =\frac{1}{2}\cdot \frac{\sqrt{3}}{2} \\
& =\frac{\sqrt{3}}{4}
\end{align}$
Thus, the left side of the expression is equal to the right side, which is $\sin {{60}^{\circ }}\sin {{30}^{\circ }}=\frac{1}{2}\left[ \cos \left( {{60}^{\circ }}-{{30}^{\circ }} \right)-\cos \left( {{60}^{\circ }}+{{30}^{\circ }} \right) \right]$.
