if tan^-1[ { (1+x²)^1/2 – (1-x²)^1/2 } / {(1+x²)^1/2 + (1-x²)^1/2 } ] = t,

then ptove that x² = sint

let tan^-1{(1+x²)^1/2- (1-x²)^1/2}/(1+x²)^1/2+(1-x²)^1/2  = t

then

[{(1+x²)^1/2 - (1-x²)^1/2} / {((1+x²)^1/2 + (1-x²)^1/2) = tan t

Now apply C and D to get,

{1-x^2/1+x²}^1/2  =  1-tant/1+tant

expand RHS by writing tan t = sint/cost.

squaring,

{1-x^2/1+x²}  = {cost - sint/cost + sint}²

{1-x^2/1+x²} = 1- sin2t/1+sin2t

so x² =sint 

Regards

Arum (askIITians forum.expert)