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

• A. $\(\frac{\sqrt2-\sqrt6}{4}\)$
• B. $\(\frac{\sqrt6-\sqrt2}{4}\)$
• C. $\(\frac{\sqrt2-\sqrt6}{2}\)$
• D. $\(\frac{\sqrt4}{4}\)$

Answer: (A)

Verified step by step guidance

1. Recognize that 105° can be expressed as the sum of two angles whose cosine values are known. Specifically, 105° = 60° + 45°.
2. Use the cosine addition formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Here, let \( a = 60° \) and \( b = 45° \).
3. Substitute the known values: \( \cos 60° = \frac{1}{2} \), \( \cos 45° = \frac{\sqrt{2}}{2} \), \( \sin 60° = \frac{\sqrt{3}}{2} \), and \( \sin 45° = \frac{\sqrt{2}}{2} \).
4. Apply the values to the formula: \( \cos 105° = \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) - \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \).
5. Simplify the expression: Calculate each term separately and then combine them to find the exact value of \( \cos 105° \).