005 Sample Final A, Question 15

Question Find an equivalent algebraic expression for the following,

$\cos(\tan ^{-1}(x))$

Foundations
1) $\tan ^{-1}(x)$ can be thought of as $\tan ^{-1}\left({\frac {x}{1}}\right),$ and this now refers to an angle in a triangle. What are the side lengths of this triangle?
Answers:
1) The side lengths are 1, x, and ${\sqrt {1+x^{2}}}.$

Step 1:
First, let $\theta =\tan ^{-1}(x)$. Then, $\tan(\theta )=x$.

Step 2:
Now, we draw the right triangle corresponding to $\theta$. Two of the side lengths are 1 and x and the hypotenuse has length ${\sqrt {x^{2}+1}}$.

Step 3:
Since $\cos(\theta )={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$, $\cos(\tan ^{-1}(x))=\cos(\theta )={\frac {1}{\sqrt {x^{2}+1}}}$.

Final Answer:
${\frac {1}{\sqrt {x^{2}+1}}}$

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