Chapter 7, Problem 12CT

To determine

To prove: The identity $\sin \theta \tan \theta +\cos \theta =\sec \theta$.

Explanation of Solution

Given information:

Instruction is to establish the identity $\sin \theta \tan \theta +\cos \theta =\sec \theta$

Formula used:

1) $\tan \theta =\frac{\sin \theta }{\cos \theta }$

2) $\sec \theta =\frac{1}{\cos \theta }$

3) Pythagorean identity: ${\sin }^2\theta +{\cos }^2\theta =1$

Proof:

Consider L.H.S. of the identity,

$\sin \theta \tan \theta +\cos \theta$

By using $\tan \theta =\frac{\sin \theta }{\cos \theta }$,

$\Rightarrow \sin \theta \cdot \frac{\sin \theta }{\cos \theta }+\cos \theta$

$=\frac{{\sin }^2\theta }{\cos \theta }+\cos \theta$

$=\frac{{\sin }^2\theta }{\cos \theta }+\frac{\cos \theta }{1}$

Multiply the numerator and denominator of $\frac{\cos \theta }{1}$ by $\cos \theta$,

$\Rightarrow \frac{{\sin }^2\theta }{\cos \theta }+\frac{{\cos }^2\theta }{\cos \theta }$

$=\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta }$

By using Pythagorean identity,

$=\frac{1}{\cos \theta }$

By using $\sec \theta =\frac{1}{\cos \theta }$

$=\sec \theta$

= Right side

Hence, the identity $\sin \theta \tan \theta +\cos \theta =\sec \theta$ is proved.