To find the exact value of $\sin(165^\circ)$ using an addition or subtraction formula, we can break $165^\circ$ into two angles whose sine and cosine values are known. For example:

$165^\circ = 180^\circ - 15^\circ$

Now, using the sine subtraction formula:

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

Substitute $A = 180^\circ$ and $B = 15^\circ$:

$\sin(165^\circ) = \sin(180^\circ - 15^\circ) = \sin 180^\circ \cos 15^\circ - \cos 180^\circ \sin 15^\circ$

We know:

$\sin 180^\circ = 0, \quad \cos 180^\circ = -1$

Thus, the formula becomes:

$\sin(165^\circ) = 0 \times \cos 15^\circ - (-1) \times \sin 15^\circ$

$\sin(165^\circ) = \sin 15^\circ$

Now, to calculate $\sin 15^\circ$, we can use the sine subtraction formula again for $15^\circ = 45^\circ - 30^\circ$:

$\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ$

From known values:

$\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}$

Substituting these values:

$\sin(15^\circ) = \left( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \times \frac{1}{2} \right)$

$\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

$\sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Therefore:

$\sin(165^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Now, regarding the equation you provided:

You mentioned an expression that looks like:
$\text{Divide}\left[\frac{\sqrt{A}}{40} \times \sqrt{B} - \frac{1}{41}, 4\right]$

This expression doesn't directly align with the standard trigonometric formulas but could potentially be a different way of writing a trigonometric expression. However, the exact value of $\sin(165^\circ)$ is:

$\sin(165^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

If you'd like to clarify or if this equation is related to something else, feel free to ask!

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Do you want more details or clarifications on any part of the solution?

Here are some follow-up questions:

1. How do you derive the sine and cosine values for specific angles like $15^\circ$?
2. What are some other useful addition and subtraction identities for trigonometric functions?
3. Can you solve trigonometric equations using these formulas?
4. How do you calculate values for trigonometric functions for non-standard angles?
5. Why does $\sin(180^\circ) = 0$ and $\cos(180^\circ) = -1$?

Tip: Using known standard angle values (like 30°, 45°, 60°) will make solving trigonometric problems easier!