We are given the equation involving trigonometric terms:

$4 \cos \theta + 4\sqrt{3} \sin \theta = 8 \sin \theta \cos \theta + 2\sqrt{3}$

We need to determine which one of the following is correct based on the equation:

(A) $\sin \theta = \frac{2}{3}$

(B) $\cos \theta = \frac{1}{2}$

(C) $\theta = 30^\circ$

(D) $\tan \theta = \sqrt{3}$

Step-by-Step Solution:

1. Simplify the given equation:

   $4 \cos \theta + 4\sqrt{3} \sin \theta = 8 \sin \theta \cos \theta + 2\sqrt{3}$

   We will try to move all the terms involving $\theta$ on one side and the constants on the other side.

2. Rearrange the equation:

   Move all terms involving $\theta$ to the left-hand side:

   $4 \cos \theta + 4\sqrt{3} \sin \theta - 8 \sin \theta \cos \theta = 2\sqrt{3}$

3. Substitute potential values:

   Let's test each option.

   • Option B: $\cos \theta = \frac{1}{2}$

     If $\cos \theta = \frac{1}{2}$, then $\theta = 60^\circ$.

     We know:

     $\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}$

     Substitute these values into the original equation:

     $4 \times \frac{1}{2} + 4\sqrt{3} \times \frac{\sqrt{3}}{2} = 8 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} + 2\sqrt{3}$

     Simplifying:

     $2 + 4 \times 3 = 4\sqrt{3} + 2\sqrt{3}$

     $2 + 12 = 6\sqrt{3}$

     But this is not a valid identity. So, Option B is incorrect.

   • Option D: $\tan \theta = \sqrt{3}$

     If $\tan \theta = \sqrt{3}$, then $\theta = 60^\circ$, which is the same as tested in Option B, and that doesn't work either.

   • Option C: $\theta = 30^\circ$

     If $\theta = 30^\circ$, then:

     $\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}$

     Substituting these values into the equation:

     $4 \times \frac{\sqrt{3}}{2} + 4\sqrt{3} \times \frac{1}{2} = 8 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} + 2\sqrt{3}$

     Simplifying:

     $2\sqrt{3} + 2\sqrt{3} = 2\sqrt{3} + 2\sqrt{3}$

     This is a valid identity. Hence, Option C is correct: $\theta = 30^\circ$.

Thus, the correct answer is:

$\boxed{\theta = 30^\circ}$

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Would you like further details or have any questions? Here are some related questions:

1. What are the values of $\sin$, $\cos$, and $\tan$ for common angles like $30^\circ$, $45^\circ$, and $60^\circ$?
2. How can you derive the trigonometric identities used in this problem?
3. What is the relationship between $\sin$ and $\cos$ for complementary angles?
4. How can you solve more complex trigonometric equations?
5. What are some common mistakes to avoid when solving trigonometric identities?

Tip: When solving trigonometric equations, always check for multiple possible solutions within the given range, especially if the problem involves acute angles or a restricted domain.