Answer
See the explanation below.
Work Step by Step
Consider the given expression $\csc x{{\cos }^{2}}x+\sin x=\csc x$.
Using the reciprocal property, we get
$\csc x=\frac{1}{\sin x}$
$\begin{align}
& \csc x{{\cos }^{2}}x+\sin x=\frac{1}{\sin x}\cdot \frac{{{\cos }^{2}}x}{1}+\sin x \\
& =\frac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\sin x}.
\end{align}$
Now, we will use the property of ${{\sin }^{2}}x+{{\cos }^{2}}x=1$.
This implies
$\begin{align}
& \csc x{{\cos }^{2}}x+\sin x=\frac{1}{\sin x} \\
& =\csc x.
\end{align}$
Hence, the provided expression is verified.
