If |x| < 1 , then prove that 2tan^-1 x = tan^-1 2x/1-x^2 = sin^-1 2x/1+x^2 = cos^-1 1-x^2/1+x^2

Question

If |x| < 1 , then prove that 

2tan^-1 x = tan^-1 \(\left(\cfrac{2x}{1-x^2}\right)\) = sin^-1 \(\left(\cfrac{2x}{1+x^2}\right)\) = cos^-1 \(\left(\cfrac{1-x^2}{1+x^2}\right)\)

2tan^-1 x = tan^-1 (2x/1-x²) = sin^-1 (2x/1+x²) = cos^-1 (1-x²/1+x²)

Answer 1

Let tan^-1 x = y 

Then, x = tan y