Answer
See the explanation below.
Work Step by Step
One of the sum-to-product formulas is $\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$. Therefore, $\sin x+\sin y$ can be written as provide below:
$\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}$
One of the sum-to-product formulas is $\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$. Therefore, $\cos x+\cos y$ can be written as shown below:
$\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$
Now, consider the left side of the provided expression:
$\frac{\sin x+\sin y}{\cos x+\cos y}$
The expression can be simplified as shown below:
$\begin{align}
& \frac{\sin x+\sin y}{\cos x+\cos y}=\frac{2\sin \left( \frac{x+y}{2} \right)\cos \left( \frac{x-y}{2} \right)}{2\cos \left( \frac{x+y}{2} \right)\cos \left( \frac{x-y}{2} \right)} \\
& =\frac{\sin \left( \frac{x+y}{2} \right)}{\cos \left( \frac{x+y}{2} \right)}
\end{align}$
Now, by using one of the quotient identities of trigonometry, which is $\tan x=\frac{\sin x}{\cos x}$, the expression can be further simplified as:
$\frac{\sin \left( \frac{x+y}{2} \right)}{\cos \left( \frac{x+y}{2} \right)}=\tan \frac{x+y}{2}$
Thus, the left side of the expression is equal to the right side, which is
$\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$.
