If alpha and beta be 2 distinct roots satisfying equation a cos theta+bsin theta=c, Show that cos (alpha+beta) = (a^2-b^2) / (a^2+b^2)

Cos (α+β) = cosα∗cosβ-sinα∗sinβ

acosθ+bsinθ=c

acosθ=c-bsinθ

Square both sides

a2 (1-sinθ) = c2+b2sin2θ-2∗c∗b∗sinθ

(b2+a2) sin2θ-2∗c∗b∗sinθ+c2-a2=0

Product of Roots = ca

sinα∗sinβ=c2-a2a2+b2

Now use similar method to have an equation in terms of cos2θ

acosθ+bsinθ=c

acosθ-c=-bsinθ

a2cos2θ+c2-2∗a∗ccosθ=b2 (1-cos2θ)

(a2+b2) cos2θ-2∗a∗cosθ∗c+c2-b2=0

Product of Roots = ca

cosα∗cosβ=c2-b2a2+b2

Now substitute the values in First Equation

cos (α+β) = cosα∗cosβ-sinα∗sinβ

⟹c2-b2+a2-c2a2+b2

⟹a2-b2a2+b2