Chapter 5.5, Problem 118E

To determine

To find:Thealgebraic expression of the trigonometric expression $\cos \left(2\arctan x\right)$ .

Answer to Problem 118E

The algebraic expression of the trigonometric expression $\cos \left(2\arctan x\right)$ is $\frac{1+x^2}{1-x^2}$ .

Explanation of Solution

Given information:

The given trigonometric expression is $\cos \left(2\arctan x\right)$ .

Calculation:

The trigonometric expression is $\cos \left(2\arctan x\right)$ .

Let,

  $\begin{array}{c}\theta =2\arctan x\\ \frac{\theta }{2}=\arctan x\\ tan\frac{\theta }{2}=x\end{array}$

Using the trigonometric formula.

  $\begin{array}{c}\cos \left(2\arctan x\right)=\cos \theta \\ ={\cos }^2\frac{\theta }{2}-{\sin }^2\frac{\theta }{2}\\ ={\cos }^2\frac{\theta }{2}\left(1-\frac{{\sin }^2\frac{\theta }{2}}{{\cos }^2\frac{\theta }{2}}\right)\\ =\frac{1}{{\sec }^2\frac{\theta }{2}}\left(1-ta{n}^2\frac{\theta }{2}\right)\end{array}$

Simplify further,

  $\begin{array}{c}\cos \left(2\arctan x\right)=\frac{\left(1-ta{n}^2\frac{\theta }{2}\right)}{1+ta{n}^2\frac{\theta }{2}}\\ =\frac{1+x^2}{1-x^2}\end{array}$

Therefore, the algebraic expression of the trigonometric expression $\cos \left(2\arctan x\right)$ is $\frac{1+x^2}{1-x^2}$ .