Chapter 14.7, Problem 22WE

To determine

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

Answer to Problem 22WE

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

Explanation of Solution

Given:

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

Calculation:

Consider the given expression is,

  $\sin \left({\cot }^{-1}x\right)$

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

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

As per trigonometric Pythagorean identity,

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

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

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