Answer
As left side transforms into right side, hence given identity-
$\tan\theta \sin \theta + \cos\theta$ = $\sec\theta$ is true.
Work Step by Step
Given identity is-
$\tan\theta \sin \theta + \cos\theta$ = $\sec\theta$
Taking L.S.
$\tan\theta \sin \theta + \cos\theta$
= $\frac{\sin\theta}{\cos \theta} . \sin\theta + \cos\theta . \frac{\cos\theta}{\cos\theta}$
( Using ratio identity for $\tan\theta$)
= $\frac{\sin^{2}\theta}{\cos \theta} + \frac{\cos^{2}\theta}{\cos\theta}$
= $\frac{\sin^{2}\theta + \cos^{2}\theta }{\cos \theta} $
= $\frac{1 }{\cos \theta} $
( Recall first Pythagorean identity, $\sin^{2}\theta + \cos^{2}\theta$ = 1)
= $\sec\theta$
= R.S.
As left side transforms into right side, hence given identity-
$\tan\theta \sin \theta + \cos\theta$ = $\sec\theta$ is true.
