Step 1: Direct check of endpoints:
At $x=0$ all terms vanish. At $x=\pi/2$: $1 - 0 + (-1) = 0$. Both satisfy the equation.
Step 2: Interior root:
Using the factored form $\sin 2x(2\cos x - 1) = 0$, the interior zero comes from $\cos x = 1/2$, giving $x = \pi/3$. Check: $\tfrac{\sqrt3}{2} - \tfrac{\sqrt3}{2} + 0 = 0$.
Step 3: Total:
There are three solutions in the closed interval: option (C).
Final Answer:
There are 3 solutions in the interval.
\[ \boxed{\text{(C) }3} \]