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