When $x=0$ then $f(x)=0$.

When $x\not = 0$ then $f(x)=\frac{x}{\sin x}$.

So, $$f(x)=\begin{cases}0 &amp; \text{ for } x=0\\\frac{x}{\sin x} &amp; \text{ for } x\in (0,\pi/2]\end{cases}$$As, $f$ is not continuous at $x=0$,  so $\{f_n(x)\}$ is NOT uniform convergent.

How to prove that $f_n(x)=\frac{nx}{1+n\sin(x)}$ does not converge uniformly on $[0, \pi/2]$?
Trying to find out which movie from approximately the 1990s contained this plot:  A woman sees a beast murder a man in an alley or street near her home.  She is terrorized and promises the beast if he lets her live she will never tell anyone about seeing the beast.  Then the movie switches from "scary" to "romantic" as she meets the man of her dreams and hooks up with him.  It appears to be a "they lived happily ever after" but she has disturbing dreams of the murder and is an artist.  She draws or sketches the beast from the alley.  Eventually she tells her hubby what she promised not to tell and he turns into the beast to destroy her.  Can't remember fully but the beast might have possibly fallen from the sky (crashing like a meteor or alien landing in the opening scene).  
