To find $\cos 225^\circ$, let's break it down:

Step 1: Locate $225^\circ$ on the unit circle

The angle $225^\circ$ is in the third quadrant. Angles in the third quadrant have the cosine value negative because the $x$-coordinate is negative.

Step 2: Reference angle

The reference angle for $225^\circ$ is: $225^\circ - 180^\circ = 45^\circ$

Step 3: Cosine of the reference angle

The cosine of $45^\circ$ is: $\cos 45^\circ = \frac{1}{\sqrt{2}}$

Step 4: Adjust for the third quadrant

Since cosine is negative in the third quadrant: $\cos 225^\circ = -\frac{1}{\sqrt{2}}$

Final Answer:

$\boxed{-\frac{1}{\sqrt{2}}}$

This matches the second option on your list.

Would you like further clarification or assistance with related trigonometric problems? Here are 5 related questions to explore:

1. What is $\sin 225^\circ$, and why is it positive in this case?
2. How do you simplify $-\frac{1}{\sqrt{2}}$ to $-\frac{\sqrt{2}}{2}$?
3. How can you use the unit circle to verify cosine values?
4. What are the cosine values for angles in all four quadrants?
5. How does the reference angle concept help with trigonometry problems?

Tip: Always remember that the sign of a trigonometric function depends on the quadrant of the angle.