To find the exact value of $\cos(285^\circ)$, let us proceed step-by-step:

Step 1: Rewrite the angle using a reference angle

The angle $285^\circ$ lies in the fourth quadrant. It can be rewritten as: $285^\circ = 360^\circ - 75^\circ$

Here, $75^\circ$ is the reference angle.

Step 2: Use the cosine difference formula

The cosine of an angle in the fourth quadrant is positive. Using the cosine difference formula: $\cos(285^\circ) = \cos(360^\circ - 75^\circ) = \cos(75^\circ)$

Step 3: Expand $\cos(75^\circ)$ using the sum formula

$\cos(75^\circ) = \cos(45^\circ + 30^\circ)$ Using the sum formula for cosine: $\cos(A + B) = \cos A \cos B - \sin A \sin B$ Substitute $A = 45^\circ$ and $B = 30^\circ$: $\cos(75^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)$

Step 4: Substitute exact trigonometric values

$\cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}$ $\sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2}$ Substitute these into the equation: $\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$

Step 5: Simplify the expression

$\cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ Combine terms: $\cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Thus: $\cos(285^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Let me know if you need further details or have questions!

Here are 5 related questions you might consider:

1. What is the exact value of $\sin(285^\circ)$?
2. How can you use the unit circle to verify $\cos(285^\circ)$?
3. What are the key trigonometric identities used in solving problems like this?
4. How does $\cos(285^\circ)$ relate to $\cos(75^\circ)$ geometrically?
5. How do you find the values of sine and cosine for non-standard angles?

Tip: Always identify the quadrant of the angle first to determine the sign of the trigonometric function.