यदि ` y= sin [ 2tan^(-1). sqrt(1-x)/sqrt(1+x)]` तो `(dy)/(dx)` का मान ज्ञात कीजिए ।

Explanation

मान लो ` x = cos theta , :. theta = cos ^(-1) x `
` y = sin [ 2 tan ^(-1). sqrt((1- cos theta)/(1+ cos theta )) ] `
` = sin [ 2 tan ^(-1) sqrt ((2sin ^(2) theta//2)/(2 cos^(2) theta //2 ))]`
` sin [ 2 tan^(-1) ( tan .theta/ 2 ) ] `
` sin (2 theta/2 ) = sin theta = sin ( cos ^(-1) x)`
` :. dy/dx = cos ( cos^(-1) x ). (-1)/(sqrt(1-x^(2)))`
` = ( -cos (cos^(-1) x))/(sqrt(1-x^(2))) " " [ :' cos ( cos^(1) theta = theta ]`
` (-x)/(sqrt(1-x^(2))). `