Question

In Problem, write each trigonometric expression as an algebraic expression in $u$.

$\tan \left({\cos }^{-1}u\right)$

Short answer

Trigonometric expression $\tan \left(co{s}^{-1}u\right)$as an algebraic expression in $u$ is given by$\begin{array}{rcl}& & \frac{\sqrt{1-u^2}}{u}\end{array}$.

Step-by-step solution

Step 1. Given information

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

Step 2. Value of the expression in terms of u

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

Let ${\cos }^{-1}u=x$

So, $u=\cos x$

Consider the following figure:

Side opposite to angle $x=\sqrt{1-u^2}$

localid="1646562158872" $\begin{array}{rcl}\tan x& =& \frac{\sqrt{1-u^2}}{u}\\ x& =& {\tan }^{-1}\left(\frac{\sqrt{1-u^2}}{u}\right)\end{array}$

Therefore,

$\begin{array}{rcl}\tan \left({\cos }^{-1}u\right)& =& \tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-u^2}}{u}\right)\right]\\ & =& \frac{\sqrt{1-u^2}}{u}\end{array}$