Let’s analyze and solve the provided expression:

The given expression is:

$\frac{\sin(x+y)}{\sin x \sin y}, \text{ with } \cot x = 15 \text{ and } \cot y = -13.$

Step 1: Using trigonometric identities

1. From $\cot x = 15$, we know: $\cot x = \frac{\cos x}{\sin x}.$ Using the Pythagorean identity: $1 + \cot^2 x = \csc^2 x.$ Substituting $\cot x = 15$: $1 + 15^2 = \csc^2 x \implies \csc^2 x = 226 \implies \sin^2 x = \frac{1}{226}.$ Hence: $\sin x = \frac{1}{\sqrt{226}}, \quad \cos x = \frac{\cot x}{\sqrt{1 + \cot^2 x}} = \frac{15}{\sqrt{226}}.$

2. For $\cot y = -13$, a similar approach: $1 + \cot^2 y = \csc^2 y.$ Substituting $\cot y = -13$: $1 + (-13)^2 = \csc^2 y \implies \csc^2 y = 170 \implies \sin^2 y = \frac{1}{170}.$ Hence: $\sin y = \frac{1}{\sqrt{170}}, \quad \cos y = \frac{\cot y}{\sqrt{1 + \cot^2 y}} = \frac{-13}{\sqrt{170}}.$

Step 2: Simplify $\sin(x + y)$

The formula for $\sin(x + y)$ is: $\sin(x + y) = \sin x \cos y + \cos x \sin y.$ Substitute the values: $\sin(x + y) = \frac{1}{\sqrt{226}} \cdot \frac{-13}{\sqrt{170}} + \frac{15}{\sqrt{226}} \cdot \frac{1}{\sqrt{170}}.$ $\sin(x + y) = \frac{-13}{\sqrt{226} \cdot \sqrt{170}} + \frac{15}{\sqrt{226} \cdot \sqrt{170}}.$ $\sin(x + y) = \frac{-13 + 15}{\sqrt{226} \cdot \sqrt{170}} = \frac{2}{\sqrt{226} \cdot \sqrt{170}}.$

Step 3: Simplify the denominator

The denominator $\sin x \sin y$ is: $\sin x \sin y = \frac{1}{\sqrt{226}} \cdot \frac{1}{\sqrt{170}} = \frac{1}{\sqrt{226} \cdot \sqrt{170}}.$

Step 4: Solve the entire expression

The entire expression becomes: $\frac{\sin(x + y)}{\sin x \sin y} = \frac{\frac{2}{\sqrt{226} \cdot \sqrt{170}}}{\frac{1}{\sqrt{226} \cdot \sqrt{170}}}.$ Simplify: $\frac{\sin(x + y)}{\sin x \sin y} = 2.$

Final Answer:

$\frac{\sin(x+y)}{\sin x \sin y} = 2.$

Let me know if you would like further clarifications or steps! Here are five related questions you might find interesting:

1. How do we simplify trigonometric expressions involving addition or subtraction?
2. What is the relationship between $\cot x$ and other trigonometric ratios?
3. How can $\sin^2 x + \cos^2 x = 1$ help simplify more complicated identities?
4. How do we determine the quadrant of angles based on trigonometric values?
5. How can this method apply to real-world problems involving angles?

Tip: Always check the quadrant of the given angles to determine the correct signs for $\sin$, $\cos$, and other trigonometric functions!