Answer
See the explanation below.
Work Step by Step
$\cot t+\frac{\sin t}{1+\cos t}=\csc t$
Recall Trigonometric Identities,
$\begin{align}
& \cot t=\frac{\cos t}{\sin t} \\
& {{\sin }^{2}}t+{{\cos }^{2}}t=1 \\
\end{align}$
Use the above identities and solve the left side of the given expression,
$\cot t+\frac{\sin t}{1+\cos t}=\csc t$
Multiply numerator and denominator of $\cot t$ by $1+\cos t$ and $\frac{\sin t}{1+\cos t}$ by $\sin t$.
$\begin{align}
& \cot t+\frac{\sin t}{1+\cos t}=\frac{\cos t}{\sin t}\left( \frac{1+\cos t}{1+\cos t} \right)+\frac{\sin t}{1+\cos t}\left( \frac{\sin t}{\sin t} \right) \\
& =\frac{\cos t+{{\cos }^{2}}t}{\sin t+\cos t\sin t}+\frac{{{\sin }^{2}}t}{\sin t+\sin t\cos t} \\
& =\frac{\cos t+1}{\sin t\left( 1+\cos t \right)} \\
& =\frac{1}{\sin t}
\end{align}$
Recall Reciprocity Identity,
$\csc t=\frac{1}{\sin t}$
Therefore,
$\cot t+\frac{\sin t}{1+\cos t}=\csc t$
