Question

Establish each identity.

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

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

Step 2. Calculation 

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

$$\begin{gathered} \frac{{\cos }^2\theta -{\sin }^2\theta }{1-{\tan }^2\theta }=\frac{{\cos }^2\theta -{\sin }^2\theta }{1-{\displaystyle \frac{{\sin }^2\theta }{{\cos }^2\theta }}} \\ =\frac{{\cos }^2\theta -{\sin }^2\theta }{{\displaystyle \frac{{\cos }^2\theta -{\sin }^2\theta }{{\cos }^2\theta }}} \\ ={\cos }^2\theta \frac{{\cos }^2\theta -{\sin }^2\theta }{{\cos }^2\theta -{\sin }^2\theta } \\ ={\cos }^2\theta \end{gathered}$$

Hence, Proved