Chapter 7, Problem 38RE

To determine

The simplified value of the expression $\cos \left({\csc }^{-1}u\right)$ in terms of $u$ .

Answer to Problem 38RE

The simplified value of the expression $\cos \left({\csc }^{-1}u\right)$ in terms of is $\frac{\sqrt{u^2-1}}{u}$ .

Explanation of Solution

Given information:

The expression is $\cos \left({\csc }^{-1}u\right)$ .

Formula used:

For the sine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Inverse of sine and inverse of cosec related as,

  ${\sin }^{-1}\left(\frac{1}{x}\right)={\csc }^{-1}x$

Where, $\left|x\right|\ge 1$ .

Cosine function and sine function related as,

  $\cos \theta =\sqrt{1-{\left(\sin \theta \right)}^2}$

Calculation:

Consider the expression $\cos \left({\csc }^{-1}u\right)$ .

Recall that inverse of sine and inverse of cosec related as,

  ${\sin }^{-1}\left(\frac{1}{x}\right)={\csc }^{-1}x$

Where, $\left|x\right|\ge 1$ .

Applied it,

  $\cos \left({\csc }^{-1}u\right)=\cos \left({\sin }^{-1}\frac{1}{u}\right)$

Where, $\left|u\right|\ge 1$ .

Again recall that cosine function and sine function related as,

  $\cos \theta =\sqrt{1-{\left(\sin \theta \right)}^2}$

Using it,

  $\cos \left({\sin }^{-1}\frac{1}{u}\right)=\sqrt{1-{\left[\sin \left({\sin }^{-1}\frac{1}{u}\right)\right]}^2}$

Where, $\left|u\right|\ge 1$ .

Observe that the value $\frac{1}{u}$ is lies between $-1$ and $1$ .

Recall that for the sine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Simplify further,

  $\begin{array}{c}\sqrt{1-{\left[\sin \left({\sin }^{-1}\frac{1}{u}\right)\right]}^2}=\sqrt{1-{\left(\frac{1}{u}\right)}^2}\\ =\frac{\sqrt{u^2-1}}{u}\end{array}$

Thus, the simplified value of the expression $\cos \left({\csc }^{-1}u\right)$ is $\frac{\sqrt{u^2-1}}{u}$ ..