Chapter 14.7, Problem 24WE

To determine

To Find: The simplified expression for the given expression without the use of trigonometric or inverse trigonometric functions.

Answer to Problem 24WE

The required expression is $\frac{x^2-1}{x^2-1}$ .

Explanation of Solution

Given:

The given expression is $\cos \left(2{\cot }^{-1}x\right)$ .

Calculation:

Consider the given expression is,

  $\cos \left(2{\cot }^{-1}x\right)$

Consider $y={\cot }^{-1}x$ , then,

  $\begin{array}{l}y={\cot }^{-1}x\\ x=\cot y\end{array}$

Consider the trigonometric identity is,

  ${\csc }^{2}y=1+{\cot }^{2}y$

As per trigonometric Pythagorean identity,

  $\begin{array}{l}\csc y=\sqrt{1+x^2}\\ \sin y=\frac{1}{\sqrt{1+x^2}}\end{array}$

Consider the expression for $\cos 2y$ is,

  $\cos 2y=1-2{\sin }^{2}y$

Then,

  $\begin{array}{c}\cos 2y=1-2{\left(\frac{1}{\sqrt{1+x^2}}\right)}^2\\ =\frac{x^2-1}{x^2-1}\end{array}$

Substitute $y={\cot }^{-1}x$ in the above equation as,

  $\cos 2\left({\cot }^{-1}x\right)=\frac{x^2-1}{x^2-1}$