Prove that : `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2,0ltxltpi/2`

Options

A. 0
B. `(1)/(2)`
C. `-(1)/(2)`
D. None of these

Answer

B

Explanation

Let ` y = cot^(-1) [(sqrt(1 + sin x)+ sqrt(1 - sin x))/(sqrt(1 + sin x)- sqrt(1 - sin x))`
`cot^(-1) [((sin""(x)/(2)+ cos""(x)/(2) ) + (cos ""(x)/(2) - sin""(x)/(2)))/((sin""(x)/(2)+ cos""(x)/(2) ) - (cos ""(x)/(2) - sin""(x)/(2)))]`
`cot^(-1) [(2cos (x//2))/(2cos (x//2))]= cot^(-1) [cot((x)/(2))]= (x)/(2)`
` rArr y = (x)/(2) rArr (dy)/(dx) = (1)/(2)`
Note (i) When ` 0 lt x lt (pi)/(2) rArr 0 (x)/(2) lt (pi)/(2)` , then
`sqrt(1 - sin x )=cos ""(x)/(2) - sin""(x)/(2)`
(ii) when `(pi)/(2) lt x lt pi, "then "sqrt(1 - sin x ) = sin"" (x)/(2)- cos""(x)/(2)`