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