If $\alpha &$ $\beta$ are the roots of the equation, $a\cos \theta+ b\sin \theta=c$ then $\sin { \alpha } +\sin { \beta } =\dfrac { 2bc }{ { a }^{ 2 }+{ b }^{ 2 } } $

The given equation is a cos x + b sin x = c. Dividing the entire equation by the square root of (a squared plus b squared), we get sin of (x plus alpha) equal to c divided by the square root of (a squared plus b squared), where tan alpha is a divided by b. For the two roots alpha and beta, we have sin alpha plus sin beta equals 2 sin of ((alpha plus beta) divided by 2) multiplied by cos of ((alpha minus beta) divided by 2). Using the sum and difference of roots, cos of ((alpha minus beta) divided by 2) equals b divided by the square root of (a squared plus b squared), and 2 sin of ((alpha plus beta) divided by 2) equals 2c divided by the square root of (a squared plus b squared). Multiplying these results gives sin alpha plus sin beta equal to 2bc divided by (a squared plus b squared), making the statement true.

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