Step 1: Check for Homogeneity: A function $f(x, y)$ is homogeneous of degree $n$ if $f(tx, ty) = t^n f(x, y)$.
Let's test our function $u(x, y)$:
$$u(tx, ty) = \sin^{-1} \left( \frac{tx}{ty} \right) + \tan^{-1} \left( \frac{ty}{tx} \right)$$
$$u(tx, ty) = \sin^{-1} \left( \frac{x}{y} \right) + \tan^{-1} \left( \frac{y}{x} \right)$$
$$u(tx, ty) = t^0 \cdot u(x, y)$$
Step 2: Identify the Degree ($n$): Since $u(tx, ty) = t^0 u(x, y)$, the function is a homogeneous function of degree $n = 0$.
Step 3: Apply Euler's Theorem: Euler's Theorem states that for a homogeneous function $u$ of degree $n$:
$$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = nu$$
Substituting our degree $n = 0$:
$$xu_x + yu_y = 0 \cdot u = 0$$
Thus, the value of the expression is always 0.