Answer
$\tan \theta\sin \theta+\cos \theta=\sec\theta$
Work Step by Step
Start with the left side:
$\tan \theta\sin \theta+\cos \theta$
Write everything in terms of cosine:
$=\frac{\sin \theta}{\cos \theta}*\sin \theta+\cos \theta$
Rewrite expression to get a common denominator:
$=\frac{\sin^2 \theta}{\cos \theta}+\frac{\cos^2\theta}{\cos \theta}$
Add the two fractions and use the identity $\sin^2\theta+\cos^2\theta=1$:
$=\frac{\sin^2 \theta+\cos^2\theta}{\cos \theta}$
$=\frac{1}{\cos \theta}$
$=\sec\theta$
Since this equals the right side, the identity has been proven.
