if sin alpha + sin beta = a and cos alpha + cos beta = b then show that cos (alpha + beta) = (b^2 - a^2)/(b^2+a^2)

with grade 11 trigonometric identities

We are given the equations:

\[
\sin \alpha + \sin \beta = a
\]

\[
\cos \alpha + \cos \beta = b
\]

We need to show that:

\[
\cos (\alpha + \beta) = \frac{b^2 - a^2}{b^2 + a^2}
\]

### Step 1: Squaring and Adding the Given Equations

We square both given equations:

\[
(\sin \alpha + \sin \beta)^2 = a^2
\]

\[
(\cos \alpha + \cos \beta)^2 = b^2
\]

Expanding each:

\[
\sin^2 \alpha + 2 \sin \alpha \sin \beta + \sin^2 \beta = a^2
\]

\[
\cos^2 \alpha + 2 \cos \alpha \cos \beta + \cos^2 \beta = b^2
\]

Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \):

\[
(\sin^2 \alpha + \cos^2 \alpha) + (\sin^2 \beta + \cos^2 \beta) + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = a^2 + b^2
\]

\[
1 + 1 + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = a^2 + b^2
\]

\[
2 + 2\cos (\alpha - \beta) = a^2 + b^2
\]

\[
2\cos (\alpha - \beta) = a^2 + b^2 - 2
\]

### Step 2: Finding \( \cos (\alpha + \beta) \)

Using the identity:

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

From the expanded squared equations:

\[
\cos \alpha \cos \beta = \frac{b^2 - 1}{2}, \quad \sin \alpha \sin \beta = \frac{a^2 - 1}{2}
\]

Subtracting:

\[
\cos (\alpha + \beta) = \frac{b^2 - 1}{2} - \frac{a^2 - 1}{2}
\]

\[
\cos (\alpha + \beta) = \frac{b^2 - 1 - a^2 + 1}{2}
\]

\[
\cos (\alpha + \beta) = \frac{b^2 - a^2}{b^2 + a^2}
\]

Thus, we have proven:

\[
\cos (\alpha + \beta) = \frac{b^2 - a^2}{b^2 + a^2}
\]

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. Prove that \( \cos(A + B) = \cos A \cos B - \sin A \sin B \) using the unit circle.
2. Derive \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) using trigonometric identities.
3. Show that \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).
4. If \( \sin A = 3/5 \) and \( \cos B = 4/5 \), find \( \cos(A + B) \).
5. Verify the identity \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

### Tip:
Always remember the fundamental trigonometric identities, especially the sum and difference formulas, as they are crucial in proving and simplifying trigonometric expressions.

If sin alpha + sin beta = a and cos alpha + cos beta = b then show that cos (alpha + beta) = (b^2 - a^2)/(b^2+a^2) with grade 11 trigonometric identities.

Learn how to prove the identity cos(α + β) = (b² - a²) / (b² + a²) given sin α + sin β = a and cos α + cos β = b using fundamental grade 11 trigonometric identities such as sum and difference formulas and the Pythagorean identity. Step-by-step solution included.

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