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Question
Prove that:
`tan^(-1) sqrtx = 1/2 cos^(-1) (1-x)/(1+x)`, x ∈ [0, 1]
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Solution
Let x = tan2 θ
⇒ `sqrtx` = tan θ
⇒ θ = `tan^(-1) sqrtx` ...(1)
∴ `(1-x)/(1+x)`
= `(1-tan^2 θ)/(1 + tan^2 θ)`
= 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) sqrtx` ....[From (1)]
R.H.S. = L.H.S.
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