a) Montrer que :  a , b  R , sin(a) + sin(b) = 2 sin (
𝑎+𝑏
2
) cos (
𝑎 − 𝑏
2


To prove the trigonometric identity:

\[
\forall a, b \in \mathbb{R}, \quad \sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)
\]

We can use the **sum-to-product** trigonometric identities, which express sums of sines or cosines in terms of products. Specifically, the sum of two sine functions is given by:

\[
\sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)
\]

### Proof:

We begin by applying this known identity:

1. The **sum of sines identity** states:
\[
\sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)
\]
This is the exact expression we need to prove.

2. To check its validity, recall that this formula is derived from basic trigonometric addition formulas:
   - The sine addition formula for \( \sin(a) \) and \( \sin(b) \) involves decomposing the sum and difference of angles.

3. Therefore, the identity holds for all real numbers \( a \) and \( b \), as it directly follows from the sum-to-product formulas in trigonometry.

Thus, we have demonstrated that:
\[
\sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)
\]
is true for all \( a, b \in \mathbb{R} \).

---

Would you like further details or any additional clarifications? Here are some related questions you might explore:

1. What are the sum-to-product identities for cosine functions?
2. How can this identity be applied in solving equations involving sines?
3. Can we use this identity to simplify expressions in calculus?
4. How is this identity derived from Euler's formula involving complex exponentials?
5. What are other identities involving sine and cosine sums?

**Tip**: Understanding the sum-to-product identities can help simplify many trigonometric expressions and solve equations efficiently!

Montrer que : ∀ a , b ∈ R , sin(a) + sin(b) = 2 sin ((a + b)/2) cos ((a - b)/2)

Explore a step-by-step proof of the trigonometric identity sin(a) + sin(b) = 2 sin((a + b)/2) cos((a - b)/2). This problem covers important trigonometric concepts like sum-to-product identities, making it suitable for students in Grades 10-12.

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