Question

Establish the identity

$\sin \theta \tan \theta +\cos \theta =sec\theta$

Short answer

$$\begin{gathered} Bywritng\tan \theta =\frac{\sin \theta }{\cos \theta } \\ weget \\ \sin \theta \tan \theta +\cos \theta =sec\theta \end{gathered}$$

Step-by-step solution

Step 1. Given information 

$\sin \theta \tan \theta +\cos \theta =sec\theta$

Step 2. Calculations

$$\begin{gathered} \sin \theta \tan \theta +\cos \theta =sec\theta \\ TakeLHS\sin \theta \tan \theta +\cos \theta \\ Weknowthat \\ \tan \theta =\frac{\sin \theta }{\cos \theta } \\ So, \\ \sin \theta \tan \theta +\cos \theta =\sin \theta \frac{\sin \theta }{\cos \theta }+\cos \theta \\ =\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta }=\frac{1}{\cos \theta }=sec\theta \\ HenceLHS=RHS \end{gathered}$$