Question

Write the expression cos(tan^-1)x) as an algebraic expression in x (without trig or inverse trig functions).

Answer 1

There is a page about those in wikipedia and you can see it in the relationship table on the bottom:
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

However, if you look at this diagram
http://en.wikipedia.org/wiki/File:Trigonometric_functions_and_inverse2.svg
It's pretty easy to see that
\[ tan(\theta) = \frac{x}{1} = x \]
Now, the hypotenuse would be:
\[ \text{hypotenuse} = \sqrt{1 + x^2} \]

Now, if we do
\[ tan^{-1}(x) = arctan(x) = \theta \]
and now we can say:
\[ cos(arctan(x)) = cos(\theta) = \frac{x}{hypotenuse} = \frac{x}{\sqrt{1+x^2}} \]