Question

Establish each identity.

$\ln |1+\cos \theta |+\ln |1-\cos \theta |=2\ln \left|\sin \theta \right|$

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$\ln |1+\cos \theta |+\ln |1-\cos \theta |=2\ln \left|\sin \theta \right|$

Step 2. Calculation 

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

$$\begin{gathered} \ln |1+\cos \theta |+\ln |1-\cos \theta |=\ln \left|\right(1+\cos \theta \left)\right(1-\cos \theta \left)\right| \\ =\ln |1-{\cos }^2\theta | \\ =\ln \left|{\sin }^2\theta \right| \\ =2\ln \left|\sin \theta \right| \end{gathered}$$

Hence, Proved