 Solution

False. If uniform convergence held, some polynomial $P_n$ would satisfy ${\sup }_{t\in \mathbb{R}}|P_n(t)-\sin t|<1$. Such a polynomial would be bounded on $\mathbb{R}$, hence constant: a nonconstant polynomial grows without bound in absolute value along a real half-line. For any constant $c$, however,a contradiction.