Question

Prove that `tan^(-1) sqrt(x) = 1/2 cos^(-1)  (1 - x)/(1 + x), x ∈ [0, 1]`.

Answer 1

Let x = tan² θ.

Then, `sqrt(x) = tan θ`

⇒ `θ = tan^(-1) sqrtx`

∴ `(1 - x)/(1 + x) = (1 - tan^2θ)/(1 + tan^2θ)`

= cos 2θ

Now, we have:

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

= `1/2 cos^(-1) (cos 2θ)`

= `1/2 xx 2θ`

= θ

= `tan^(-1) sqrt(x)`

= L.H.S.