JEE Math Practice Question

Let $g$ be a differentiable function such that $\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0$ and let $y=y(x)$ satisfy the differential equation $\frac{d y}{d x}-y \tan x=2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\right)$. If $y(0)=0$, then $y\left(\frac{\pi}{3}\right)$ is equal to

1. A.$\frac{4 \pi}{3}$
2. B.$\frac{2 \pi}{3}$
3. C.$\frac{2 \pi}{3 \sqrt{3}}$
4. D.$\frac{4 \pi}{3 \sqrt{3}}$