To determine the exact value of $\cos 465^\circ$, we can reduce the angle to a standard position within the range of $0^\circ$ to $360^\circ$ by subtracting $360^\circ$ from $465^\circ$:

$465^\circ - 360^\circ = 105^\circ$

Thus, $\cos 465^\circ = \cos 105^\circ$.

Now, we need to determine the value of $\cos 105^\circ$. Notice that $105^\circ$ is in the second quadrant, where the cosine function is negative. Additionally, we can express $105^\circ$ as:

$105^\circ = 180^\circ - 75^\circ$

Using the identity for cosine in the second quadrant, $\cos(180^\circ - \theta) = -\cos \theta$, we get:

$\cos 105^\circ = -\cos 75^\circ$

The cosine of $75^\circ$ can be calculated using the angle sum identity:

$\cos(75^\circ) = \cos(45^\circ + 30^\circ)$

Using the formula for the cosine of a sum:

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

Substitute $a = 45^\circ$ and $b = 30^\circ$:

$\cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$

Using the known 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 into the equation:

$\cos 75^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)$

$\cos 75^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Therefore:

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

Thus, the exact value of $\cos 465^\circ$ is:

$\cos 465^\circ = -\frac{\sqrt{6} - \sqrt{2}}{4}$

Would you like more details on any step or have any questions?

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Here are some related questions:

1. How do you find exact trigonometric values of other angles outside $0^\circ$ to $360^\circ$?
2. What is the sine of $465^\circ$?
3. How does the cosine function behave in different quadrants?
4. Can you use the angle difference identity for similar problems?
5. What are some other trigonometric identities involving cosine?

Tip: Remember that trigonometric functions repeat their values every $360^\circ$, which is called periodicity.