Find the exact value of the expression.
$\(\tan\)105\(\degree\)$

• A. 0
• B. $\(\sqrt\)3-2$
• C. $-2-\(\sqrt\)3$
• D. $\(\frac{\sqrt3+2}{2}\)$

Answer: (C)

Verified step by step guidance

1. First, recognize that the angle 105° can be expressed as the sum of two angles whose tangent values are known: 105° = 60° + 45°.
2. Use the tangent addition formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). Here, \( a = 60° \) and \( b = 45° \).
3. Substitute the known values: \( \tan 60° = \sqrt{3} \) and \( \tan 45° = 1 \) into the formula.
4. Calculate the numerator: \( \tan 60° + \tan 45° = \sqrt{3} + 1 \).
5. Calculate the denominator: \( 1 - \tan 60° \tan 45° = 1 - \sqrt{3} \times 1 = 1 - \sqrt{3} \).