Consider a curve $\gamma(t) = (x_{1}(t),x_{2}(t))$, such as : 
Suppose angle between parts of the curve is $\alpha \ne \pi/2$. Is it possible to represent it like a smooth curve $(x_{1}(t),x_{2}(t))$,where $x_{i}(t) \in C^{\infty}$?
I know that it's possible for $\alpha = \pi/2$, but what about this curve?
Yes, that's possible. Just let the speed of the curve slow down in a smooth fashion to $0$ when approaching the corner (use something which behaves like $\exp(-1/x^2)$ near $x=0$).
But not as a regular curve, i.e. a curve with nonvanishing derivative.