Answer
See the full explanation below.
Work Step by Step
Let us consider the left side of the given expression:
$\tan \left( \theta +\frac{\pi }{4} \right)$
By using the identities of trigonometry,
$\sin \,\left( \alpha +\beta \right)=\sin \,\alpha \,\cos \beta +\cos \,\alpha \,\sin \,\beta $
$\cos \,\left( \alpha +\beta \right)=\cos \,\alpha \,\cos \beta -\sin \,\alpha \,\sin \,\beta $ and $\tan \,\alpha =\frac{\sin \,\alpha }{\cos \,\alpha }$ , the above expression can be further simplified as:
$\begin{align}
& \tan \left( \theta +\frac{\pi }{4} \right)=\frac{\sin \left( \theta +\frac{\pi }{4} \right)}{\cos \left( \theta +\frac{\pi }{4} \right)} \\
& =\frac{\sin \,\theta \,\cos \frac{\pi }{4}+\cos \,\theta \,\sin \,\frac{\pi }{4}}{\cos \,\theta \,\cos \frac{\pi }{4}-\sin \,\theta \,\sin \,\frac{\pi }{4}} \\
& =\frac{\sin \,\theta \times \frac{1}{\sqrt{2}}+\cos \,\theta \times \frac{1}{\sqrt{2}}}{\cos \,\theta \times \frac{1}{\sqrt{2}}-\sin \,\theta \times \frac{1}{\sqrt{2}}} \\
& =\frac{\frac{1}{\sqrt{2}}\times \left( \sin \,\theta +\cos \,\theta \right)}{\frac{1}{\sqrt{2}}\times \left( \cos \,\theta -\sin \,\theta \right)}
\end{align}$
$=\frac{\cos \,\theta +\sin \,\theta }{\cos \,\theta -\sin \,\theta }$
Hence, the the left side of the given expression is equal to the right side, which is $\tan \left( \theta +\frac{\pi }{4} \right)=\frac{\cos \,\theta +\sin \,\theta }{\cos \,\theta -\sin \,\theta }$.
