Question

if alpha and beta are two distinct roots of acos theta + bsin theta = c find cos(alpha-beta)

Answer 1

  a tanX+bsecX=c................(i)
 a tanX-c=-b secX
(a tanX-c)²=(-bsecX)²
a² tan²X+c²-2actanx=b²sec²X
a²tan²X + c²-2ac tanX=b²(1+tan²X)
a²tan²X(a²-b²)-2ac tan X=b²+b² tan²X
tan²X(a²-b²)-2ac tanX+(c²-b²)=0
Hence,tanα+tanβ=2ac/(a²-b²)
tanα tanβ=c²-b²/a²-b²
tan(α+β)=tanα+tanβ/1-tanα tanβ=2ac/(a²-b²)÷c²-b²/(a²-b²)=2ac/(a²-b²)÷a²-c²/a²-b²=2ac/(a²-c²)