How do we derive the addition formula of $\sin u$ from the following equation?
$$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$
Motivation
Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - t^2}}$ Then $x = \sin u$
Let $v = \int_{0}^{y}\frac{dt}{\sqrt{1 - t^2}}$ Then $y = \sin v$
Let $u + v = const.$
Then $d(u + v) = \frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$