Question

Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`

Answer 1

L.H.S. = `1/2"cos"^-1((1-"x")/(1+"x"))`

Put x = tan²θ ⇒ tan θ = `sqrt"x" => theta = "tan"^-1 sqrt"x"` 

`= 1/2 "cos"^-1 ((1 - "tan"^2theta)/(1 + "tan"^2theta)) = 1/2 "cos"^-1 ("cos"2theta)`

`= 1/2 xx 2theta = theta = "tan"^-1 sqrt"x" = "R.H.S"`