Calculate: $\;$ $\cos \left[2 \tan^{-1} \left(2\right)\right]$
Given Expression (GE): $\;$ $\cos \left[2 \tan^{-1} \left(2\right)\right]$
$= 2 \times \cos^2 \left(\tan^{-1} 2\right) - 1$
$= 2 \left[\cos \left(\tan^{-1} 2\right)\right]^2 - 1$
$= 2 \times \left(\dfrac{1}{\sqrt{1 + 2^2}}\right) - 1$
$= 2 \times \left(\dfrac{1}{\sqrt{5}}\right)^2 - 1 = \dfrac{2}{5} - 1 = \dfrac{-3}{5}$
Formulas used:
$\cos 2 \theta = 2 \cos^2 \theta - 1$
$\cos \left(\tan^{-1} x\right) = \dfrac{1}{\sqrt{1 + x^2}}, \;\; 0 \leq x \leq 1$