To find $\cos(s + t)$ and $\cos(s - t)$, we will use the cosine sum and difference identities.

The identities are as follows:

$\cos(s + t) = \cos(s) \cos(t) - \sin(s) \sin(t)$ $\cos(s - t) = \cos(s) \cos(t) + \sin(s) \sin(t)$

We are given:

• $\sin(s) = -\frac{4}{5}$ with $s$ in Quadrant III
• $\sin(t) = \frac{12}{13}$ with $t$ in Quadrant II

Step 1: Find $\cos(s)$ and $\cos(t)$

For $\cos(s)$:

Since $s$ is in Quadrant III, both $\sin(s)$ and $\cos(s)$ are negative.

Using the Pythagorean identity: $\sin^2(s) + \cos^2(s) = 1$ $\left( -\frac{4}{5} \right)^2 + \cos^2(s) = 1$ $\frac{16}{25} + \cos^2(s) = 1$ $\cos^2(s) = 1 - \frac{16}{25} = \frac{9}{25}$ $\cos(s) = -\frac{3}{5}$

For $\cos(t)$:

Since $t$ is in Quadrant II, $\sin(t)$ is positive and $\cos(t)$ is negative.

Using the Pythagorean identity: $\sin^2(t) + \cos^2(t) = 1$ $\left( \frac{12}{13} \right)^2 + \cos^2(t) = 1$ $\frac{144}{169} + \cos^2(t) = 1$ $\cos^2(t) = 1 - \frac{144}{169} = \frac{25}{169}$ $\cos(t) = -\frac{5}{13}$

Step 2: Substitute into the Cosine Sum and Difference Identities

For $\cos(s + t)$:

$\cos(s + t) = \cos(s) \cos(t) - \sin(s) \sin(t)$ Substitute the values: $\cos(s + t) = \left( -\frac{3}{5} \right) \left( -\frac{5}{13} \right) - \left( -\frac{4}{5} \right) \left( \frac{12}{13} \right)$ $= \frac{15}{65} + \frac{48}{65} = \frac{63}{65}$

For $\cos(s - t)$:

$\cos(s - t) = \cos(s) \cos(t) + \sin(s) \sin(t)$ Substitute the values: $\cos(s - t) = \left( -\frac{3}{5} \right) \left( -\frac{5}{13} \right) + \left( -\frac{4}{5} \right) \left( \frac{12}{13} \right)$ $= \frac{15}{65} - \frac{48}{65} = -\frac{33}{65}$

Final Answers:

$\cos(s + t) = \frac{63}{65}$ $\cos(s - t) = -\frac{33}{65}$

Would you like more details or further assistance?

Here are related questions for further exploration:

1. How would these results change if $s$ and $t$ were in different quadrants?
2. Can we solve for $\tan(s + t)$ using similar identities?
3. What are the identities for $\sin(s + t)$ and $\sin(s - t)$?
4. How do the quadrant signs affect the result of trigonometric identities?
5. How would we apply these identities if $\sin(s)$ and $\sin(t)$ were in decimal form?

Tip: Memorizing quadrant sign conventions for trigonometric functions helps solve problems quickly and accurately.