Let $\alpha$ and $\beta$ be two distinct roots of $a\, \cos\theta+b\, \sin\,\theta=c$. where a, b, c are three real constants and $\theta\in [0, 2\pi]$. Then, $\alpha+\beta$ is also a root of the same equation, if

We are given that a cos(alpha) + b sin(alpha) = c, a cos(beta) + b sin(beta) = c, and a cos(alpha + beta) + b sin(alpha + beta) = c. Squaring and adding the first two equations gives a^2 + b^2 + 2ab sin(alpha + beta) = 2c^2. Expanding the third equation squared yields a^2 cos^2(alpha + beta) + b^2 sin^2(alpha + beta) + 2ab sin(alpha + beta) = c^2. Equating the expressions, a^2 + b^2 + 2ab sin(alpha + beta) = 2a^2 cos^2(alpha + beta) + 2b^2 sin^2(alpha + beta) + 4ab sin(alpha + beta). This equality holds for arbitrary angles if and only if a = c, leading to the condition c = a.

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