Answer
Showed that given statement, $\tan\theta + \cot\theta$ = $\sec\theta\csc\theta$,
is an identity as left side transforms into right side.
Work Step by Step
Given statement is-
$\tan\theta + \cot\theta$ = $\sec\theta\csc\theta$
Left Side = $\tan\theta + \cot\theta$
= $\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}$
(Using ratio identity for $\tan\theta$ and $\cot\theta$)
=$\frac{\sin\theta}{\sin\theta}.\frac{\sin\theta}{\cos\theta} +\frac{\cos\theta}{\sin\theta}. \frac{\cos\theta}{\cos\theta} $
=$\frac{\sin^{2}\theta}{\sin\theta\cos\theta} + \frac{\cos^{2}\theta}{\sin\theta\cos\theta} $
=$\frac{\sin^{2}\theta + \cos^{2}\theta}{\sin\theta\cos\theta}$
=$\frac{1}{\sin\theta\cos\theta}$
=$\frac{1}{\cos\theta}.\frac{1}{\sin\theta}$
= $\sec\theta\csc\theta$
= Right Side
i.e. Left Side transforms into Right Side
i.e. Given statement, $\tan\theta + \cot\theta$ = $\sec\theta\csc\theta$,
is an identity.
