Q 2 ii Prove: tan-11+x-1-x1+x+1-x=π4-12cos-1x, 0 x 1 - Maths - Inverse Trigonometric Functions

Question

Q   2   i i   P r o v e :   tan - 1 1 + x - 1 -
x 1 + x + 1 - x = π 4 - 1 2 cos - 1 x ,   0 < x < 1

Lovina Kansal · Accepted Answer

Dear student
tan-11+x-1-x1+x+1-xPutting x=cos2θ,we havetan-11+cos2θ-1-cos2θ1+cos2θ+1-cos2θ=tan-12cos2θ-2sin2θ2cos2θ+2sin2θ      cos2A=1-2sin2A=2cos2A-1 and ∵0<x<1⇒0<cos2θ<1⇒⇒0<2θ<π2⇒0<θ<π4=tan-1cosθ-sinθcosθ+sinθ=tan-11-tanθ1+tanθ=tan-1tanπ4-θ=π4-θ     ∵0<θ<π4⇒0<π4-θ<π4=π4-12cos-1x    ∵cos2θ=x ∴2θ=cos-1x⇒θ=12cos-1x
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Q 2 i i P r o v e : tan - 1 1 + x - 1 - x 1 + x + 1 - x = π 4 - 1 2 cos - 1 x , 0 < x < 1

$Q 2 (i i) P r o v e : \tan^{- 1} \{\frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}\} = \frac{π}{4} - \frac{1}{2} \cos^{- 1} x, [0 < x < 1]$