Answer
See the explanation below.
Work Step by Step
To verify the given identity,
$\tan \theta +\cot \theta =\sec \theta \csc \theta $
Recall Trigonometric Identities,
$\begin{align}
& {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\
& \tan \theta =\frac{\sin \theta }{\cos \theta } \\
& \cot \theta =\frac{\cos \theta }{\sin \theta } \\
\end{align}$
Use the above identities and solve the left side of the given expression,
$\begin{align}
& \tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\
& =\frac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{\sin \theta \cos \theta } \\
& =\frac{1}{\sin \theta \cos \theta } \\
& =\frac{1}{\sin \theta }\left( \frac{1}{\cos \theta } \right)
\end{align}$
Recall Reciprocal Identities,
$\begin{align}
& \csc \theta =\frac{1}{\sin \theta } \\
& \sec \theta =\frac{1}{\cos \theta } \\
\end{align}$
Therefore,
$\tan \theta +\cot \theta =\sec \theta \csc \theta $
Hence, it is proved that the given identity holds true.
