Answer
$\dfrac{\cos{\theta}+1}{\sin{\theta}+\tan{\theta}}=\cot{\theta}$
Work Step by Step
Use agraphing utility to graph $y=\dfrac{\cos{\theta}+1}{\sin{\theta}+\tan{\theta}}$.
Refer to the graph below.
Notice that the graph looks the same as that of $y=\cot{\theta}$.
RECALL:
$\tan{\theta}=\dfrac{\sin{\theta}}{\cos{\theta}}\\\\$
Use the definition above to obtain:
\begin{align*}
\frac{\cos{\theta}+1}{\sin{\theta}+\tan{\theta}}&=\frac{\cos{\theta}+1}{\sin{\theta}+\frac{\sin{\theta}}{\cos{\theta}}}\\\\
&=\frac{\cos{\theta}+1}{\frac{\sin{\theta}\cos{\theta}}{\cos{\theta}}+\frac{\sin{\theta}}{\cos{\theta}}}\\\\
&=\frac{\cos{\theta}+1}{\frac{\sin{\theta}\cos{\theta}+\sin{\theta}}{\cos{\theta}}}\\\\
&=\frac{\cos{\theta}+1}{\frac{\sin{\theta}(\cos{\theta}+1)}{\cos{\theta}}}\\\\
&=(\cos{\theta}+1) \cdot \frac{\cos{\theta}}{\sin{\theta}(\cos{\theta}+1)}\\\\
&=\frac{\cos{\theta}}{\sin{\theta}}\\\\
&=\cot{\theta}
\end{align*}
Therefore,
$$\dfrac{\cos{\theta}+1}{\sin{\theta}+\tan{\theta}}=\cot{\theta}$$
