To find the area of the shaded region, we need to calculate the area of triangle ABC and subtract the area of sector ADE.

The area of triangle ABC is given by:

\(\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 4 \times 4 \tan \alpha = 8 \tan \alpha\)

The area of sector ADE is given by:

\(\text{Area of sector } ADE = \frac{1}{2} \times 2^2 \times \alpha = 2 \alpha\)

Therefore, the area of the shaded region is:

\(8 \tan \alpha - 2 \alpha\)