if alpha and beta are the distinct roots of acostheta+bsintheta=c then prove that cos (alpha+ beta) = (a²-b²)/(a²+b²)

We are given the equation \( a \cos \theta + b \sin \theta = c \), and we know that \(\alpha\) and \(\beta\) are the distinct roots of this equation. We are asked to prove that:

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

### Step 1: Using the sum of angles formula

We know the following

If alpha and beta are the distinct roots of a cos(theta) + b sin(theta) = c, then prove that cos(alpha + beta) = (a² - b²)/(a² + b²).

Explore the solution to a trigonometric equation with distinct roots alpha and beta, and learn how to prove that cos(alpha + beta) equals (a² - b²)/(a² + b²). This problem covers concepts in trigonometry and angle sum identities, suitable for students in Grades 11-12.

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Solution

Step 1: Using the sum of angles formula

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