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To compute \(\cos 195^\circ\), we can use the angle subtraction identity. Specifically:

\[
\cos(195^\circ) = \cos(180^\circ + 15^\circ)
\]

Using the identity \(\cos(180^\circ + \theta) = -\cos(\theta)\), we get:

\[
\cos(195^\circ) = -\cos(15^\circ)
\]

Next, we use the cosine of the sum identity for \(15^\circ = 45^\circ - 30^\circ\):

\[
\cos(15^\circ) = \cos(45^\circ - 30^\circ)
\]

The identity for cosine of a difference is:

\[
\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)
\]

So:

\[
\cos(15^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ)
\]

Using the exact values:

\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2}
\]

Substituting these values:

\[
\cos(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
\]

\[
\cos(15^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

Thus:

\[
\cos(195^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{4}
\]

The correct answer is the first option:

\[
-\frac{\sqrt{2} + \sqrt{6}}{4}
\]

Would you like more details or have any questions? Here are some follow-up questions to consider:

1. How do you derive the identities for cosine of sum and difference?
2. What are the exact values of trigonometric functions at common angles (e.g., \(30^\circ\), \(45^\circ\), \(60^\circ\))?
3. Can you compute \(\sin(195^\circ)\) using a similar approach?
4. What is the significance of the unit circle in determining the values of trigonometric functions?
5. How does the symmetry of the cosine function affect its values for angles greater than \(180^\circ\)?

**Tip:** Always double-check angle identities and their corresponding values, especially when dealing with trigonometric functions in different quadrants.

Learn how to compute cos 195° by using angle subtraction and exact trigonometric values. This problem explores trigonometric identities, cosine of sum and difference, and involves detailed steps to derive the final value. Suitable for students in Grades 10-12.

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