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Prove `Cot^(-1) ((Sqrt(1+Sin X) + Sqrt(1-sinx))/(Sqrt(1+Sin X) - Sqrt(1- Sinx))) = X/2`, `X in (0, Pi/4)` - Mathematics

Prove `cot^(-1)  ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = x/2`, `x in (0, pi/4)` 

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Solution

Consider `((sqrt(1+sinx) + sqrt(1-sin x))/(sqrt(1+sinx) - sqrt(1-sinx))) = x/2` `x in (0, pi/4)`

`= ((sqrt(1+sinx)+ sqrt(1-sinx))^2)/((sqrt(1+sin x))^2 - (sqrt(1-sin x))^2)`            (by rationalizing)

`= ((1+sinx) + (1-sinx)+2sqrt((1+sinx)(1-sinx)))/(1+sinx - 1 + sinx)`

`=(2(1+sqrt(1-sin^2x)))/(2sin x) = (1+ cosx)/sin x = (2 cos^2  x/2)/(2sin  x/2 cos  x/2)`

`= cot x/2`

:. L.H.S = `cot^(-1)  ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = cot^(-1) (cot x/2) = x/2 =  R.H.S`

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APPEARS IN

NCERT Class 12 Maths
Chapter 2 Inverse Trigonometric Functions
Q 10 | Page 52
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