\(\arcsin (-1) = -\frac{\pi}{2}\)
\(\arcsin \frac{1}{2} = \frac{\pi}{6}\)
\(\operatorname{tg} \left(-\frac{\pi}{2} + \frac{\pi}{6}\right) = \operatorname{tg} \left(-\frac{3\pi}{6} + \frac{\pi}{6}\right) = \operatorname{tg} \left(-\frac{2\pi}{6}\right) = \operatorname{tg} \left(-\frac{\pi}{3}\right) = -\sqrt{3}\)