What is the value of $\cos\left(2\tan^{-1} \frac{1}{2}\right)$?

Explanation

1. 01

   Let $\theta = \tan^{-1} \frac{1}{2}$: opposite 1, adjacent 2, hypotenuse $\sqrt{5}$, so $\cos \theta = \frac{2}{\sqrt{5}}$.

2. 02

   $\cos 2\theta = 2\cos^{2} \theta - 1 = \frac{8}{5} - 1 = \frac{3}{5}$.