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

Options

A. `sqrt(1-x^(2)) `
B. `(1)/(sqrt(1-x^(2)))`
C. `(1)/(2sqrt(1-x^(2)))`
D. x

Answer

C

Explanation

Let ` y = tan ^ ( -1 ) (( sqrt( 1 + x ) - sqrt ( 1 + x )) /( sqrt ( 1 + x ) + sqrt ( 1 - x )) ) `
Put ` x = cos 2 theta `
` therefore y = tan ^( -1 ) `
` (( sqrt( 1 + cos 2 theta) - sqrt ( 1 - cos 2 theta))/( sqrt ( 1 + cos 2 theta ) + sqrt ( 1 -cos theta)) ) `
` = tan ^( - 1) (( sqrt2 cos theta - sqrt 2 sin theta)/( sqrt 2 cos theta + sqrt 2 sin theta )) `
= ` tan ^( -1) (( 1 - tan theta ) /( 1 + tan theta) ) `
` = tan ^( - 1 ) { tan ((pi ) /(4 ) - theta ) } `
` = (pi ) /( 4 ) - ( 1 )/( 2 ) cos ^( -1 ) x `
On differentiating W.r.t x we get
` (dy ) / (dx ) = 0 + ( 1 ) /( 2 ) * ( 1 ) /(sqrt ( 1 - x ^2 )) = ( 1 ) /( 2sqrt ( 1 - x ^ 2 )) `