Step 1: Identify the form of the differential equation.

The given equation is:

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

This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$, where $P = 2 \tan x$ and $Q = \sin x$.

Step 2: Find the Integrating Factor (I.F.).

$$I.F. = e^{\int P \, dx} = e^{\int 2 \tan x \, dx} = e^{2 \log |\sec x|} = e^{\log (\sec^2 x)} = \sec^2 x$$

Step 3: Write the general solution.

The solution is given by:

$$y \cdot (I.F.) = \int (Q \cdot I.F.) \, dx + C$$

Substituting the values, we get:

$$y \sec^2 x = \int (\sin x \cdot \sec^2 x) \, dx + C$$ $$y \sec^2 x = \int (\tan x \sec x) \, dx + C$$ $$y \sec^2 x = \sec x + C$$

Step 4: Find the particular solution using the given condition.

Given condition: $y = 0$ when $x = \frac{\pi}{3}$.

Substituting these values into the general solution:

$$0 \cdot \sec^2 \left(\frac{\pi}{3}\right) = \sec \left(\frac{\pi}{3}\right) + C$$ $$0 = 2 + C$$ $$C = -2$$

Step 5: Write the final particular solution.

Substituting $C = -2$ into the general solution:

$$y \sec^2 x = \sec x - 2$$ $$y = \frac{\sec x - 2}{\sec^2 x}$$ $$y = \cos x - 2\cos^2 x$$