y=coth^-1(sinx)

Question

find dy/dx and simplify the answer

Answer 1

coth(y)=1/tanh(y)=cosh(y)/sinh(y)=(e^y+e^-y)/(e^y-e^-y).

d(coth(y))/dy=[(e^y-e^-y)(e^y-e^-y)-(e^y+e^-y)(e^y+e^-y)]/(e^y-e^-y)²=

1-(e^y+e^-y)²/(e^y-e^-y)²=1-coth²(y).

If coth(y)=sin(x):

d(coth(y))/dx=d(coth(y))/dy.(dy/dx)=(1-coth²(y))dy/dx=cos(x),

dy/dx=cos(x)/(1-sin²(x))=cos(x)/cos²(x)=1/cos(x)=sec(x), because coth(y)=sin(x).