Question

How do you find the exact value of `cos(tan^-1 2)`?

Answer 1

A tangent of `2/1` is opposite `2`, adjacent `1` so hypotenuse `sqrt{2^2+1^2}=sqrt{5},` so cosine of `pm 1/sqrt{5}` or if we mean the principal value

`cos( text{Arc}text{tan}(2)) =1/sqrt{5}`

Explanation:

Stop using the `tan^-1(x)` notation -- `arctan(x)` is better, and is the official standard.