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Question
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
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Solution
Let `tan^-1 sqrt"x"` = y
∴ tan y = `sqrt"x"`
∴ x = tan2y
Now,
RHS = `1/2 cos^-1 ((1 - "x")/(1 + "x"))`
`= 1/2 cos^-1 ((1 - tan^2 "y")/(1 + tan^2 "y"))`
`= 1/2 cos^-1 (cos 2"y")`
`= 1/2 (2"y") = "y"`
`= tan^-1 sqrt"x"`
= LHS.
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