Uniform continuity of $f(x)=(1-\cos(x))/\sin(x)$ on the interval $(0,1)$

Question

I got this question:

Prove that the function $f(x)=\frac{1-\cos(x)}{\sin(x)}$ is uniformly continuous on the interval $(0,1)$

I tried to prove it directly using the definition of uniform continuity but I failed this way. Then I tried to prove it using the fact that the sum of two uniformly continuous functions is uniformly continuous by writing $f(x)=\frac{1}{\sin(x)} - \cot(x)$ and then I tried to show that both $\frac{1}{\sin(x)}$ and $cot(x)$ are uniformly continuous on (0,1) but I wasn't managed to proceed that much.

Some hints will be helpful. Thanks.

Answer 1

Hint: You can extend $f$ continuously to $[0,1]$ and continuous functions on closed intervals are uniformly continuous.

Answer 2

Hint: $$\frac{1-\cos x}{\sin x}\cdot\frac{1+\cos x}{1+\cos x}=\cdots$$

Answer 3

Extension to Simon's answer: $$\lim\limits_{x\to 0}{\frac{1-\cos\,x}{\sin\,x}}=\lim_{x\to 0}{\frac{\frac{1}{2}x^2}{x}}=0$$ the limit exists, so you can extend $f$ from $(0,1)$ to $[0,1]$