Given that, #y=sin^4(cot^-1sqrt{(1-x)/(1+x)}),# we need #dy/dx.#
We substitute #x=cos2theta," so that, "-1 lt x lt 1.#
Note that, in order to make #sqrt((1-x)/(1+x))# meaningful, we must have
#-1 lt x lt 1,# which justifies our substitution : #x=cos2theta.#
#:. y=sin^4(cot^-1sqrt{(1-x)/(1+x)}),#
#=sin^4(cot^-1sqrt{(1-cos2theta)/(1+cos2theta)}),#
#=sin^4(cot^-1sqrt{(2sin^2theta)/(2cos^2theta)}),#
#=sin^4(cot^-1(tantheta)),#
#=sin^4(cot^-1{cot(pi/2-theta)}),#
#=sin^4(pi/2-theta),#
#={sin(pi/2-theta)}^4,#
#=cos^4theta,#
#=(cos^2theta)^2,#
#={(1+cos2theta)/2}^2.#
# rArr y=1/4(1+x)^2........................................[because, cos2theta=x].#
#:. dy/dx=1/4*d/dx(1+x)^2,#
#=1/4*2(1+x)*d/dx(1+x)..............[because," the Chain Rule],"#
# rArr dy/dx=1/2(1+x).#
Enjoy Maths.!
