Find the function that is implicitly defined in the relation

Question

We are given a relation $\sqrt{ x^{2} - y^{2}} + \cos^{-1} (\frac{x}{y}) = 0$ and are asked to find a function $g(x)$ that is implicitly defined within it. $y = x$ works but how can I show this is the only function that works and if there are more how can I find them?

Answer 1

The main branches of $\sqrt{\cdot}$ and $\cos^{-1}$ have nonnegative ranges, so their sum is zero only if they are both zero themselves. This forces $y = x$.