Question

How do you evaluate `cos(tan^-1(1/2))` without a calculator?

Answer 1

`+-2/sqrt5`

Explanation:

let `tan^-1(1/2) =a => tan a=1/2`

`H=sqrt(2^2+1^2)=sqrt5`

`cos(tan^-1(1/2))=cos a = 2/H=2/sqrt5`

As tangent is positive in both the first and the third quadrants,
`=> cosa=+-2/sqrt5`

positive sign for `a` in the first quadrant and negative sign for `a` in the third quadrant.