Which of these functions are strictly increasing on $\left(0, \frac{\pi}{2}\right)$?

I. $\sin x$

II. $\cos x$

III. $\tan x$

IV. $\sin x - x$

Explanation

1. 01

   $\cos x > 0$ and $\sec^{2} x > 0$ there, so $\sin x$ and $\tan x$ rise; $\cos x$ falls, as $-\sin x < 0$.

2. 02

   $\frac{d}{dx}(\sin x - x) = \cos x - 1 < 0$ on $\left(0, \frac{\pi}{2}\right)$, so IV falls.