Answer
The statement is true.
Work Step by Step
First we will find the derivative using the Chain Rule:
$$g'(x)=(f(x)\sin x)'=f'(x)\sin x+f(x)(\sin x)'=f'(x)\sin x+f(x)\cos x$$
Now we will evaluate $g'(x)$ for $x=0$:
$$g'(0)=f'(0)\sin0+f(0)\cos0=f'(0)\cdot0+f(0)\cdot1=f(0)$$
So, the statement is true.
