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Prove that tan^(-1) [(√(1+x)-√(1-x))/(√(1+x)+√(1-x))]=pi/4-1/2 cos^(-1)x,-1/√2<=x<=1 - Mathematics

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Prove that

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`

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Solution

To prove

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`

Taking LHS, we get:

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]`

let `x=cos 2theta`

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]=tan^(-1) [(sqrt(1+cos2theta)-sqrt(1-cos2theta))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]`

`=tan^(-1)[(costheta-sintheta)/(costheta+sintheta)]`

`=tan^(-1)[(1-tantheta)/(1+tantheta)]`

`=tan^(-1) tan(pi/4-theta)`

`=(pi/4-theta)`

`=π/4−θ`

`=π/4−1/2cos^(−1) x`

`=RHS       `

Hence proved.

Concept: Inverse Trigonometric Functions (Simplification and Examples)
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