The image shows a trigonometric identity to prove:

$\sin(\alpha + \beta) \sin(\alpha - \beta) = \sin^2(\alpha) - \sin^2(\beta)$

To prove this identity, we can use the following sum and difference formulas for sine:

• $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$
• $\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)$

Step-by-Step Proof:

1. Expand both sine terms: $\sin(\alpha + \beta) \sin(\alpha - \beta) = (\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)) (\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta))$

2. Use the distributive property (FOIL method): $= \sin(\alpha)\cos(\beta) \cdot \sin(\alpha)\cos(\beta) - \sin(\alpha)\cos(\beta) \cdot \cos(\alpha)\sin(\beta) + \cos(\alpha)\sin(\beta) \cdot \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \cdot \cos(\alpha)\sin(\beta)$

3. Simplify each term: $= \sin^2(\alpha)\cos^2(\beta) - \sin(\alpha)\cos(\beta)\cos(\alpha)\sin(\beta) + \cos(\alpha)\sin(\beta)\sin(\alpha)\cos(\beta) - \cos^2(\alpha)\sin^2(\beta)$

4. Combine like terms (notice the middle terms cancel out): $= \sin^2(\alpha)\cos^2(\beta) - \cos^2(\alpha)\sin^2(\beta)$

5. Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ for each sine and cosine term to express this equation in the form of the required identity: $= \sin^2(\alpha) - \sin^2(\beta)$

Thus, we have proven that: $\sin(\alpha + \beta) \sin(\alpha - \beta) = \sin^2(\alpha) - \sin^2(\beta)$

Let me know if you need more details or further clarification!

Here are some additional questions to explore related concepts:

1. How would you prove the identity for $\cos(\alpha + \beta)\cos(\alpha - \beta)$?
2. What is the general method for proving trigonometric identities?
3. Can this proof be generalized to other trigonometric functions, like tangent or secant?
4. How can we use this identity to solve trigonometric equations?
5. What role does the sum-to-product identity play in simplifying trigonometric expressions?

Tip: When dealing with complex trigonometric identities, breaking the expression into smaller parts and applying basic sum/difference formulas can often make the simplification process easier.