Choosing signs in inverse trigonometric composing

Question

I understood why he chose the positive square root in the sin but why the tan is also positive ? Isn't the tan positive and negative in this interval ?

Answer 1

Note that when $-1 <x < 0$, $$\tan(\cos^{-1}(x)) = \frac{\sqrt{1-x^2}}{x} < 0.$$

Answer 2

Note that, for $\theta\in[0,\pi]$ and $\theta\ne\pi/2$, $$ \tan\theta=\frac{\sin\theta}{\cos\theta} $$ Since you have already established that $\sin\arccos x=\sqrt{1-x^2}$, you can directly conclude that $$ \tan\arccos x=\frac{\sqrt{1-x^2}}{x} $$ because $\cos\arccos x=x$ by definition.

Using $\pm$ is misleading, in my opinion.