Chapter 14.7, Problem 12WE

To determine

To Find: The value of the given trigonometric equation.

Answer to Problem 12WE

The value of the expression is $\frac{5}{12}$ .

Explanation of Solution

Given:

The given expression is $\cot \left[{\sin }^{-1}\frac{12}{13}\right]$ .

Calculation:

Consider the given trigonometric expression is,

  $\cot \left[{\sin }^{-1}\frac{12}{13}\right]$

Consider for $\left|x\right|\le 1$ the value of ${\sin }^{-1}\left(x\right)={\csc }^{-1}\left(\frac{1}{x}\right)$ then,

  ${\sin }^{-1}\left(\frac{12}{13}\right)={\csc }^{-1}\left(\frac{13}{12}\right)$

Consider $x={\csc }^{-1}\left(\frac{13}{12}\right)$ then,

  $\csc x=\left(\frac{13}{12}\right)$

Then,

  $\begin{array}{l}\cot x=\sqrt{{\left(\frac{13}{12}\right)}^2-1}\\ \cot \left({\sin }^{-1}\frac{12}{13}\right)=\frac{5}{12}\end{array}$