If x = cot(y) what is dy/dx?

Here we will use:

cot(x) = cos(x)/sin(x)

cosec²(x) = 1 + cot²(x)

Chain rule : dy/dt * dt/ dx = dy/ dx

Product rule : d/dx (uv) = udv/dx + v*du/dx

x = cot(y)= cos(y)/sin(y)

dx/dy = -cos(y)(^cos(y)/_sin(y)²) + 1/sin(y) (-sin(y))

         = -cos²(y)/sin²(y) -1 = - cos²(y)/sin²(y) -sin²(y)​ /sin²(y)

         = -(sin²(y)+ cos²(y))/sin²(y)​ = -cosec²(y)

cosec²(y) = 1 + cot²(y)

x = cot(y)

dx/dy = -(1 + x²)

dy/dx = -1/(1+x²)