Explanation

(b)
Since, $u$ is not a homogeneous function. But $f(x,y) = \sin{u = \frac{x^{2} + y^{2}}{x + y}}\$is a homogeneous function of degree one.
Here, by Euler’s theorem,
$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = f$
$\Rightarrow x\frac{\partial}{\partial x}(\sin{u) + y\frac{\partial}{\partial y}(\sin{u) = \sin u}}$
$\Rightarrow \ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = \tan u$