Answer
The result of $\frac{\cos \left( \alpha -\beta \right)+\cos \left( \alpha +\beta \right)}{-\sin \left( \alpha -\beta \right)+\sin \left( \alpha +\beta \right)}$ is $\cot \beta $.
Work Step by Step
$\frac{\cos \left( \alpha -\beta \right)+\cos \left( \alpha +\beta \right)}{-\sin \left( \alpha -\beta \right)+\sin \left( \alpha +\beta \right)}$
by using the trigonometric identities,
$\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta $ , $\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta $
$\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta $ , $\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta $
Now, the above expression can be further simplified as,
$\begin{align}
& \frac{\cos \left( \alpha -\beta \right)+\cos \left( \alpha +\beta \right)}{-\sin \left( \alpha -\beta \right)+\sin \left( \alpha +\beta \right)}=\frac{\left( \cos \alpha \cos \beta +\sin \alpha \sin \beta \right)+\left( \cos \alpha \cos \beta -\sin \alpha \sin \beta \right)}{-\left( \sin \alpha \cos \beta -\cos \alpha \sin \beta \right)+\left( \sin \alpha \cos \beta +\cos \alpha \sin \beta \right)} \\
& =\frac{\cos \alpha \cos \beta +\sin \alpha \sin \beta +\cos \alpha \cos \beta -\sin \alpha \sin \beta }{-\sin \alpha \cos \beta +\cos \alpha \sin \beta +\sin \alpha \cos \beta +\cos \alpha \sin \beta } \\
& =\frac{\cos \alpha \cos \beta +\cos \alpha \cos \beta }{\cos \alpha \sin \beta +\cos \alpha \sin \beta } \\
& =\frac{2\cos \alpha \cos \beta }{2\cos \alpha \sin \beta }
\end{align}$
Use, $\frac{\cos x}{\sin x}=\cot x$
$\begin{align}
& \frac{2\cos \alpha \cos \beta }{2\cos \alpha \sin \beta }=\frac{\cos \beta }{\sin \beta } \\
& =\cot \beta
\end{align}$
Thus, $\frac{\cos \left( \alpha -\beta \right)+\cos \left( \alpha +\beta \right)}{-\sin \left( \alpha -\beta \right)+\sin \left( \alpha +\beta \right)}$ can be simplified as $\cot \beta $.
Hence, the result of $\frac{\cos \left( \alpha -\beta \right)+\cos \left( \alpha +\beta \right)}{-\sin \left( \alpha -\beta \right)+\sin \left( \alpha +\beta \right)}$ is $\cot \beta $.
