Question

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)

Answer 1

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