Use an addition or subtraction formula to find the exact value of the expression.
\[
\tan \left(255^{\circ}\right)
\]
a. $1+\sqrt{3}$
b. $1+\frac{1}{\sqrt{3}}$
c. $2+\sqrt{3}$
d. $2+\frac{1}{\sqrt{3}}$

Step 1 ：Given the angle is 255 degrees, we can express this angle as a difference of two angles whose tangent values we know. The angles 180 degrees and 75 degrees are such angles.

Step 2 ：We know that \(\tan(180^{\circ}) = 0\) and \(\tan(75^{\circ})\) can be expressed as \(\tan(45^{\circ} + 30^{\circ})\).

Step 3 ：We can use the formula for the tangent of the sum of two angles, which is \(\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\), to find the value of \(\tan(75^{\circ})\).

Step 4 ：Then we can use the formula for the tangent of the difference of two angles, which is \(\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}\), to find the value of \(\tan(255^{\circ})\).

Step 5 ：The value of \(\tan(255^{\circ})\) is approximately -3.732, which is the negative of the value of \(\tan(75^{\circ})\). This makes sense because the tangent function has a period of 180 degrees, so the tangent of an angle and the tangent of that angle plus 180 degrees are negatives of each other.

Step 6 ：The exact value of \(\tan(75^{\circ})\) is \(2 + \sqrt{3}\), so the exact value of \(\tan(255^{\circ})\) is \(-(2 + \sqrt{3})\). This is not one of the given options, so there may be a mistake in the problem or in my calculations.

Step 7 ：\(\boxed{\text{Final Answer: None of the given options is correct. The exact value of } \tan(255^{\circ}) \text{ is } -(2 + \sqrt{3})}\)