Answer
See the full explanation below.
Work Step by Step
Let us consider the left side of the given expression:
$\sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)$
Now, evaluate the expression and substitute the appropriate values:
$\begin{align}
& \sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)=\sin {{90}^{{}^\circ }} \\
& =1
\end{align}$
And consider the right side of the provided expression:
$\sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}$
Now, evaluate the expression and substitute the appropriate values:
$\begin{align}
& \sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}=\frac{1}{2}+\frac{\sqrt{3}}{2} \\
& =\frac{1+\sqrt{3}}{2}
\end{align}$
Thus, the left side of the given expression is not equal to the right side, which is $\sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)$ or $\sin {{90}^{{}^\circ }}$ and equal to $\sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}$.
Hence, they are not equal.
(b)
Let us consider the left side of the given expression:
$\sin \left( 30{}^\circ +60{}^\circ \right)$
Now, evaluate the expression and substitute the appropriate values:
$\begin{align}
& \sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)=\sin {{90}^{{}^\circ }} \\
& =1
\end{align}$
Now, consider the right side of the provided expression:
$\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ $
And, evaluate the expression and substitute the appropriate values:
$\begin{align}
& \sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ ~=\left( \frac{1}{2} \right)\left( \frac{1}{2} \right)+\left( \frac{\sqrt{3}}{2} \right)\left( \frac{\sqrt{3}}{2} \right) \\
& =\frac{1}{4}+\frac{3}{4} \\
& =1
\end{align}$
Thus, the left side of the given expression is equal to the right side, which is $\sin \left( 30{}^\circ +60{}^\circ \right)$ or $\sin 90{}^\circ $ and equal to $\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ $.
Hence, they are equal.
