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Prove that Prove that 1/2 "Cos"^(-1) ((1-"X")/(1+"X")) = "Tan"^-1 Sqrt"X" - Mathematics

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Sum

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

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Solution

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"`

Concept: Concept of Continuity
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