Let y=y(x) be the solution of the differential

$\frac{d y}{d x}+2 y \tan x=2 \sin x$

I.F. $=e^{\int 2 \tan x d x}=\sec ^{2} x$

The solution of the differential equation is

$y \times$ I.F. $=\int$ I.F $\times 2 \sin x d x+C$

$\Rightarrow y \cdot \sec ^{2} x=\int 2 \sin x \cdot \sec ^{2} x d x+C$

$\Rightarrow y \sec ^{2} x=2 \sec x+C$......(1)

When $x=\frac{\pi}{3}, y=0 ;$ then $C=-4$

$\therefore$ From (1), $y \sec ^{2} x=2 \sec x-4$

$\Rightarrow y=\frac{2 \sec x-4}{\sec ^{2} x} \Rightarrow y\left(\frac{\pi}{4}\right)=\sqrt{2}-2$