Chapter 7.2, Problem 57AYU

In Problems 57-66, write each trigonometric expression as an algebraic expression in $u$.

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

To determine

To find: Trigonometric expression as algebraic expression in $u$.

Answer to Problem 57AYU

Trigonometric expression $=\text{ cos}\left({\text{tan}}^{-1}u\right) = \text{algebraic expression}\hspace{0.17em}=\hspace{0.17em}1/\surd \left(1 + u^2\right)$.

Explanation of Solution

Given:

Trigonometric expression $= \text{cos}\left({\text{tan}}^{-1}u\right)$.

Formula used:

$\text{Tan}\theta =1/\text{cot}\theta$

Calculation:

Let $\theta = {\text{tan}}^{-1}u$.   -----1

$\text{tan}\theta = u$; $-\pi /2<\theta <\pi /2$ and $-\infty < u < \infty$

Therefore $\text{cot }\theta =1/u$.   -----2

From equation 1

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

Multipling by 1,

$=\text{ cos}\theta =\text{ cos}\theta x1$

Multiplying and dividing by $\text{sin}\theta$.

$=\text{ cos}x\left(\text{sin}\theta /\text{sin}\theta \right)$

$\left(\text{cos}\theta /\text{sin}\theta \right)x\text{sin}\theta$

Using the formula,

$\text{sin}\theta =1/\text{cosec}\theta$

$\text{cot}\theta x\left(1/\text{cosec}\theta \right)$

Using the Formula,

$\text{cosec}\theta =\surd \left(1 + {\text{cot}}^2\right)$

$\text{cot}\theta x1/\surd \left(1 + {\text{cot}}^2\right)$

Using equation 2

$\left(1/u\right) x1/\surd \left(1 + 1/u^2\right)$

$\left(1/u\right) x \left(u/\left(\surd u^2 + 1\right)\right)$

Trigonometric expression $= \text{cos}\left({\text{tan}}^{-1}u\right)=1/\surd \left(1 + u^2\right)$.