`(dy)/(dx) ` का मान ज्ञात कीजिए|
`(i) y=tan ^(-1) {sqrt( 1+cos x ) /( 1-cos x )}`
` y= tan ^(-1) { (1+ sin x )/(1-sin x ) } `

Explanation

(i ) हम जानते है
`y= tan ^(-1) { (1+cos x) /( 1-cos x ) } `
`= tan ^(-1) { sqrt(( 2cos ^(2) (x//2)) /( 2sin ^(2)(x//2)))}`
` = tan ^(-1) (cot ""(x)/(2) ) =tan ^(-1) {tan ""(( pi)/(2) -(x)/(2))}`
` " "y= ((pi)/(2)- (x)/(2))`
` therefore " "(dy)/(dx) =(d)/(dx) ((pi )/(2) -(x)/(2)) =(d)/(dx) ((pi)/( 2)) -(d)/(dx) ((x)/(2))`
` " "=(0) -((1)/(2) )=- (1)/(2)`
(ii) हम जानते है ` y= tan^(-1) {sqrt((1+sin x )/( 1- sin x )}} ` ltbr gt` =tan ^(-1) {(1-cos ""((pi )/(2) +x) )/( 1+ cos""((pi)/(2) +x) ) } ^(1//2)`
` " "=tan ^(-1) { (2sin ^(2) ""((pi )/(4) + (x)/(2) ) )/( 2cos""(( pi)/(4) +(x)/(2) ))} ^(1//2)`
` " "=tan ^(-1) { tan ""( (pi)/(4) + (x)/(2))}= ((pi)/(4)+(x)/(2)) `
` therefore " "y= ((pi )/(4) +(x)/(2) ) `
`therefore " "(dy)/(dx) = (d)/(dx) ((pi )/(4) + (x)/(2))`
` rArr " "(d)/(dx) ((pi)/(4)) +(d)/(dx) ((x)/(2)) =0 + (1)/(2) = (1)/(2) `