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The problem you’ve uploaded asks to prove two trigonometric identities for sums and differences of angles involving sine.

### Identity 1: 
\[
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]

### Proof:
We start by using the angle addition formula for sine:

\[
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]

This is the standard angle addition formula for sine and holds true by definition. Thus, the identity is directly proven.

### Identity 2:
\[
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
\]

### Proof:
For this identity, we use the angle subtraction formula for sine:

\[
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
\]

Again, this is the standard sine subtraction formula, and the identity holds by definition.

Both identities are derived from basic trigonometric angle addition and subtraction formulas. 

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Would you like further details or have any questions?

Here are some related questions to explore the topic further:

1. How does the cosine addition formula compare to the sine addition formula?
2. Can you derive the cosine subtraction formula in a similar way?
3. How can the addition and subtraction formulas be used to simplify trigonometric expressions?
4. What is the significance of these formulas in solving trigonometric equations?
5. How can these formulas help in solving real-world problems involving angles?

**Tip:** Always remember to look for patterns in trigonometric identities—they often have symmetrical structures that simplify the memorization process.

Prove that sin(α + β) = sin α cos β + cos α sin β, and sin(α − β) = sin α cos β − cos α sin β.

Explore how to prove the sine angle addition and subtraction formulas: sin(α + β) = sin α cos β + cos α sin β, and sin(α − β) = sin α cos β − cos α sin β. This problem is suitable for students in Grades 10-12 studying trigonometry.

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