Question

Find a formula for cos(tan−1(2x)) in terms of x

Answer 1

$~\cos\left( \tan^{-1} 2x\right)\\\\\\=\cos \left[\cos^{-1} \dfrac{1}{\sqrt{(2x)^2+1^2}} \right]~~~~~~~~~~~~~~~~~~~~~~~~~~;[\text{Pythagorean theorem}]\\\\\\=\cos \left( \cos^{-1} \dfrac{1}{\sqrt{4x^2 +1}} \right)\\\\\\=\dfrac{1}{\sqrt{4x^2 +1}}$