1. We are asked to evaluate the definite integral $$\int_0^{\pi/3} 2 \sec^{2} x \, dx$$. 2. Recall that the integral of $\sec^{2} x$ with respect to $x$ is $\tan x$: $$\int \sec^{2} x \, dx = \tan x + C$$. 3. Use the constant multiple rule for integrals: $$\int 2 \sec^{2} x \, dx = 2 \int \sec^{2} x \, dx = 2 \tan x + C$$. 4. Now, apply the definite integral limits from $0$ to $\frac{\pi}{3}$: $$\int_0^{\pi/3} 2 \sec^{2} x \, dx = \left[ 2 \tan x \right]_0^{\pi/3} = 2 \tan \frac{\pi}{3} - 2 \tan 0$$. 5. Evaluate the tangent values: $$\tan \frac{\pi}{3} = \sqrt{3}, \quad \tan 0 = 0$$. 6. Substitute these values back in: $$2 \sqrt{3} - 2 \times 0 = 2 \sqrt{3}$$. 7. Therefore, the value of the definite integral is $$\boxed{2 \sqrt{3}}$$.