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