Chapter 14.7, Problem 19WE

To determine

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

Answer to Problem 19WE

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

Explanation of Solution

Given:

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

Calculation:

Consider the given expression is,

  $\csc \left({\sin }^{-1}x\right)$

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

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

Then, the given equation is,

  $\csc \left(y\right)=\frac{1}{\sin y}$

Substitute $x=\sin y$ in the above equation as,

  $\csc \left(y\right)=\frac{1}{x}$