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Prove `Tan^(-1) ((Sqrt(1+X) - Sqrt(1-x))/(Sqrt(1+X) + Sqrt(1-x))) = Pi/4 - 1/2 Cos^(-1) X. - 1/Sqrt(2) <= X <= 1` [Hint: Put X = `Cos 2 Theta`] - CBSE (Commerce) Class 12 - Mathematics

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Question

Prove `tan^(-1)  ((sqrt(1+x) - sqrt(1-x))/(sqrt(1+x) + sqrt(1-x))) = pi/4 - 1/2 cos^(-1) x. - 1/sqrt(2) <= x <= 1` [Hint: put x =  `cos 2 theta`]

Solution

Put x = `cos 2theta` so that `theta = 1/2 cos^(-1) x` Then we have

L.H.S =` tan^(-1)  ((sqrt(1+x) - sqrt(1-x))/(sqrt(1+ x) + sqrt(1-x)))`

=` tan^(-1)  ((sqrt(1+cos2theta) - sqrt(1-cos 2theta))/(sqrt(1+cos2theta) + sqrt(1-cos 2 theta)))`

`= tan^(-1)  ((sqrt(2cos^2theta) -  sqrt(2sin^2theta))/(sqrt(2cos^2 theta)+sqrt(2 sin^2 theta)))`

`tan^(-1)  ((sqrt2 cos theta - sqrt2 sin theta)/(sqrt2 cos theta + sqrt2 sin theta))`

`= tan^(-1)  ((cos theta - sin theta)/(cos theta + sin theta)) = tan^(-1)  ((1 - tan^ theta)/(1+tan theta))`

`= tan^(-1) 1 - tan^(-1) (tan theta)`    `"    "[tan^(-1) ((x-y)/(1+xy))= tan^(-1) x -  tan^(-1) y] `

`= pi/4 -  theta = pi/4 -  1/2 cos^(-1) x` =R.H.S 

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APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 2: Inverse Trigonometric Functions
Q: 11 | Page no. 52
Solution Prove `Tan^(-1) ((Sqrt(1+X) - Sqrt(1-x))/(Sqrt(1+X) + Sqrt(1-x))) = Pi/4 - 1/2 Cos^(-1) X. - 1/Sqrt(2) <= X <= 1` [Hint: Put X = `Cos 2 Theta`] Concept: Properties of Inverse Trigonometric Functions.
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