Chọn phương án C.

Theo công thức hạ bậc ta có $$\begin{aligned}
4\sin^2x=3\Leftrightarrow&4\cdot\dfrac{1-\cos2x}{2}=3\\
\Leftrightarrow&2-2\cos2x=3\\
\Leftrightarrow&\cos2x=-\dfrac{1}{2}\\
\Leftrightarrow&\left[\begin{array}{l}2x=\dfrac{2\pi}{3}+k2\pi\\ 2x=-\dfrac{2\pi}{3}+k2\pi\end{array}\right.\\
\Leftrightarrow&\left[\begin{array}{l}x=\dfrac{\pi}{3}+k\pi\\ x=-\dfrac{\pi}{3}+k\pi\end{array}\right.\,(k\in\mathbb{Z})
\end{aligned}$$