Implicit differentiation and inverse trigonometric functions practice question 13

Give the inner angle a name. What is its tangent, and in which range of angles must it lie?

Let $\theta = \arctan(-2)$, so $\tan\theta = -2$ with $-\tfrac{\pi}{2} < \theta < \tfrac{\pi}{2}$. Picture a right triangle with opposite side $2$ and adjacent side $1$.

The hypotenuse is $\sqrt{5}$. Cosine is positive for every angle strictly between $-\tfrac{\pi}{2}$ and $\tfrac{\pi}{2}$, which settles the sign.