Question

Establish each identity.

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

Short answer

The left side of the given equation has been simplified to bring the right side.

Step-by-step solution

Step 1. Given Information

The given equation is$\ln \left|sec\theta \right|=-\ln \left|\cos \theta \right|$

Step 2. Calculation 

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$$\begin{gathered} \ln \left|sec\theta \right|=\ln \left|\frac{1}{\cos \theta }\right| \\ =\ln |\cos \theta {|}^{-1} \\ =-\ln |\cos \theta | \end{gathered}$$

Hence, Proved