Solve: sin⁻¹(x) + sin⁻¹(1-x) = cos⁻¹ x.

$\sin^{-1}(x) + \sin^{-1}(1-x) = \cos^{-1} x$ $\Rightarrow \sin^{-1}(x) + \sin^{-1}(1-x) = \frac{\pi}{2} - \sin^{-1} x$ $\Rightarrow \sin^{-1}(1-x) = \frac{\pi}{2} - 2\sin^{-1} x$ $\Rightarrow (1-x) = \sin\left(\frac{\pi}{2} - 2\sin^{-1} x\right)$ $\Rightarrow (1-x) = \cos(2\sin^{-1} x)$ $\Rightarrow (1-x) = \cos(\cos^{-1}(1 - 2x^2))$ $\Rightarrow (1-x) = 1 - 2x^2$ $\Rightarrow 2x^2 - x = 0$ $\therefore x = 0, \frac{1}{2}$