Answer
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Work Step by Step
Considered the identity, $\tan \theta +\cot \theta =\sec \theta \csc \theta $
Now, apply the Quotient identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\cot \theta =\frac{\cos \theta }{\sin \theta }$ ,
$\begin{align}
& \tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\
& =\frac{\sin \theta \cdot \sin \theta +\cos \theta \cdot \cos \theta }{\sin \theta \cos \theta } \\
& =\frac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{\sin \theta \cos \theta }
\end{align}$
Then, use the Pythagorean identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ ,
$\tan \theta +\cot \theta =\frac{1}{\sin \theta \cos \theta }$
Now, use the Reciprocal identity $\csc \theta =\frac{1}{\sin \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$ ,
$\begin{align}
& \tan \theta +\cot \theta =\left( \frac{1}{\cos \theta } \right)\left( \frac{1}{\sin \theta } \right) \\
& =\sec \theta \csc \theta
\end{align}$
Hence, the left side is identical to the right side $\tan \theta +\cot \theta =\sec \theta \csc \theta $.
