Differentiate `tan^(-1){sqrt((1+sinx)/(1-sinx))}`,` -pi/2ltxltpi/2` with respect to `x` :

Explanation

Given that,
`y=tan^(-1){sqrt((1+sinx)/(1-sinx))}`,
` -pi/2ltxltpi/2`
`y=tan^-1 sqrt[(sin^2(x/2)+cos^2(x/2)+2sin(x/2)cos(x/2))/(sin^2(x/2)+cos^2(x/2) -2 sin(x/2) cos(x/2))]`
`=>y=tan^-1sqrt[(sin(x/2)+cos(x/2))^2/(cos(x/2)-sin(x/2))^2]`
`=>y=tan^-1[(sin(x/2)+cos(x/2))/(cos(x/2)-sin(x/2))]`
`=>y=tan^-1[(sin(x/2)+cos(x/2))/cos(x/2)/(cos(x/2)-sin(x/2))/cos(x/2)]`
`=>y=tan^-1[(1+tan(x/2))/(1-tan(x/2))]`
`=>y=tan^-1[(tan(pi/4)+tan(x/2))/(1-tan(x/2)*tan(pi/4))]`
`=>y=tan^-1 [tan(pi/4+x/2)]`
` -pi/4 lt x/2 lt pi/4`
`=>0 lt pi/4+x/2 lt pi/2`
`y= pi/4+x/2`
`(dy)/(dx)=1/2`