To find the exact value of $\cos(105^\circ)$, we can use the cosine addition formula. We know that:

$\cos(a + b) = \cos a \cos b - \sin a \sin b$

Since $105^\circ = 60^\circ + 45^\circ$, we can apply the formula as follows:

$\cos(105^\circ) = \cos(60^\circ + 45^\circ)$

Step-by-Step Solution:

1. Apply the Cosine Addition Formula: $\cos(105^\circ) = \cos(60^\circ) \cos(45^\circ) - \sin(60^\circ) \sin(45^\circ)$

2. Substitute Known Values:

   • $\cos(60^\circ) = \frac{1}{2}$
   • $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
   • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
   • $\sin(45^\circ) = \frac{\sqrt{2}}{2}$

   Substituting these values gives: $\cos(105^\circ) = \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) - \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right)$

3. Simplify the Expression: $\cos(105^\circ) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$

4. Combine Terms: $\cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4}$

Final Answer:

$\cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4}$

Would you like further details on the steps?

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Here are five related questions to deepen your understanding:

1. How would you use the sine addition formula to find $\sin(105^\circ)$?
2. What is the exact value of $\cos(75^\circ)$ using the same approach?
3. How can the cosine subtraction formula be used to calculate $\cos(15^\circ)$?
4. How does the value of $\cos(105^\circ)$ compare to $\cos(75^\circ)$?
5. Why does breaking down angles into known values help in finding exact trigonometric values?

Tip: Remember, addition and subtraction formulas are especially helpful for angles that aren't common in the unit circle but can be decomposed into known angles.