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