Question
3. Prove the following: a) \( \sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \)
Ask by Estrada Bob. in South Africa
Jan 12,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}
\]
Solution
To prove the identity
\[
\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2},
\]
we can use the sine subtraction formula. The sine subtraction formula states that:
\[
\sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right).
\]
In our case, we can let \( A = x \) and \( B = y \). Therefore, we can rewrite the left-hand side:
\[
\sin x - \sin y = 2 \cos \left( \frac{x+y}{2} \right) \sin \left( \frac{x-y}{2} \right).
\]
Now, let's break it down step by step:
1. **Identify \( A \) and \( B \)**:
- Let \( A = x \)
- Let \( B = y \)
2. **Apply the sine subtraction formula**:
- According to the formula, we have:
\[
\sin x - \sin y = 2 \cos \left( \frac{x+y}{2} \right) \sin \left( \frac{x-y}{2} \right).
\]
3. **Conclusion**:
- Since we have derived the right-hand side from the left-hand side using the sine subtraction formula, we have proven that:
\[
\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}.
\]
Thus, the identity is proven.
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Simplify this solution The Deep Dive
To prove \( \sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \), we can use the sine subtraction formula. Recall that the sine difference can be expressed using the formula: \[ \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \] In this case, let \( A = x \) and \( B = y \). Therefore, we have: \[ \sin x - \sin y = 2 \cos \left( \frac{x+y}{2} \right) \sin \left( \frac{x-y}{2} \right) \] This matches the given expression, thereby proving that \( \sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \). Thus, the proof is complete!
