Homogeneous function: sin u=(√x−√y)/(√x+√y)

Given sin u = (√x − √y)/(√x + √y). Let

z = sin u = (x^(1/2) − y^(1/2))/(x^(1/2) + y^(1/2)).

Homogeneity

Replace x, y by tx, ty. The factor t^(1/2) cancels between numerator and denominator:

z(tx, ty) = t^(1/2)(√x − √y)/(t^(1/2)(√x + √y)) = t⁰ z.

So z is a homogeneous function of degree n = 0.

Euler's theorem

For a homogeneous function of degree n: x ∂z/∂x + y ∂z/∂y = n z. Here n = 0:

x ∂z/∂x + y ∂z/∂y = 0

Since z = sin u, ∂z/∂x = cos u ∂u/∂x and ∂z/∂y = cos u ∂u/∂y:

cos u ( x ∂u/∂x + y ∂u/∂y ) = 0
x ∂u/∂x + y ∂u/∂y = 0      (cos u ≠ 0)

Hence proved. ∎