Chapter 9.7, Problem 40E

a.

To determine

To verify: If $\ln \left|\sec \theta \right|=-\ln \left|\cos \theta \right|$

Explanation of Solution

Given information:

Given −

  $\ln \left|\sec \theta \right|=-\ln \left|\cos \theta \right|$ .

Calculation:

Given −

  $\ln \left|\sec \theta \right|=-\ln \left|\cos \theta \right|$ .

  $\begin{array}{l}\text{Now let us verify the identity,}\\ \text{LHS}=\ln \left|\sec \theta \right|\\ =\ln \left|\frac{1}{\cos \theta }\right|\left\{\text{Since, }\sec \theta =\frac{1}{\cos \theta }\right\}\\ =\ln \left|1\right|-\ln \left|\cos \theta \right|\left\{\text{Since, }\ln \left(\frac{a}{b}\right)=\ln a-\ln b\right\}\\ =-\ln \left|\cos \theta \right|\left\{\text{Since, }\ln 1=0\right\}\\ =\text{RHS}\end{array}$

  $$

Hence, $\ln \left|\sec \theta \right|=-\ln \left|\cos \theta \right|$ .

b.

To determine

To verify: If $\ln \left|\tan \theta \right|=\ln \left|\sin \theta \right|-\ln \left|\cos \theta \right|$

Explanation of Solution

Given information:

Given −

  $\ln \left|\tan \theta \right|=\ln \left|\sin \theta \right|-\ln \left|\cos \theta \right|$ .

Calculation:

Given −

  $\ln \left|\tan \theta \right|=\ln \left|\sin \theta \right|-\ln \left|\cos \theta \right|$ .

  $\begin{array}{l}\text{Now let us verify the identity,}\\ \text{LHS}=\ln \left|\tan \theta \right|\\ =\ln \left|\frac{\sin \theta }{\cos \theta }\right|\left\{\text{Since, }\tan \theta =\frac{\sin \theta }{\cos \theta }\right\}\\ =\ln \left|\sin \theta \right|-\ln \left|\cos \theta \right|\left\{\text{Since, }\ln \left(\frac{a}{b}\right)=\ln a-\ln b\right\}\\ =\text{RHS}\end{array}$

Hence, $\ln \left|\tan \theta \right|=\ln \left|\sin \theta \right|-\ln \left|\cos \theta \right|$ .