Chapter 13.3, Problem 48SR

To determine

To verify: The given identity.

Explanation of Solution

Given:

The identity is given as:

  $\sec \theta \left(\sec \theta -\cos \theta \right)={\tan }^2\theta$

According to the reciprocal identity:

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

Now, considering the left hand side of the identity:

  $\begin{array}{c}\sec \theta \left(\sec \theta -\cos \theta \right)=\frac{1}{\cos \theta }\left(\frac{1}{\cos \theta }-\cos \theta \right)\\ =\frac{1}{\cos \theta }\left(\frac{1-{\cos }^2\theta }{\cos \theta }\right)\\ =\left(\frac{1-{\cos }^2\theta }{{\cos }^2\theta }\right)\mathrm{......}\left(1\right)\end{array}$

From the Pythagoras identity:

  $\begin{array}{l}{\sin }^2\theta +{\cos }^2\theta =1\\ {\sin }^2\theta =1-{\cos }^2\theta \mathrm{........}\left(2\right)\end{array}$

From equation 1 and 2:

  $\sec \theta \left(\sec \theta -\cos \theta \right)=\frac{{\sin }^2\theta }{{\cos }^2\theta }$

According to the quotient identity:

  $\begin{array}{l}\frac{\sin \theta }{\cos \theta }=\tan \theta \\ \frac{{\sin }^2\theta }{{\cos }^2\theta }={\tan }^2\theta \end{array}$

Hence,

  $\sec \theta \left(\sec \theta -\cos \theta \right)={\tan }^2\theta$

From the above result it is clear that the left hand side of the identity is equal to the right hand side of the identity.

Therefore, the identity is verified.