Which of the following functions are increasing on $x ∈(0,\frac{\pi}{2})$?

(A) $f(x) = \sin x$
(B) $f(x) = \cos x$
(C) $f(x) = \tan x$
(D) $f(x) = \cos 3x$

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (2) → (A) and (C) only

(A) $f(x) = \sin x$
(C) $f(x) = \tan x$

Given interval: $x \in (0, \frac{\pi}{2})$

Check monotonicity using derivative:

(A) $f(x) = \sin x$

$f'(x) = \cos x > 0$ in $(0, \frac{\pi}{2})$

⇒ Increasing

(B) $f(x) = \cos x$

$f'(x) = -\sin x < 0$ in $(0, \frac{\pi}{2})$

⇒ Decreasing

(C) $f(x) = \tan x$

$f'(x) = \sec^2 x > 0$ in $(0, \frac{\pi}{2})$

⇒ Increasing

(D) $f(x) = \cos 3x$

$f'(x) = -3\sin 3x < 0$ in $(0, \frac{\pi}{2})$

⇒ Decreasing