if alpha and beta are roots of a cos theta + b sin theta=c,show that cos(alpha+beta)=a^2-b^2/a^2+b^2.
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Step-by-step explanation:
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It is given that α and β are distinct roots of acosx+bsinx=c.
Therefore,
acosα+bsinα=c and acosβ+bsinβ=c
⇒(acosα+bsinα)−(acosβ+bsinβ)=c−c
⇒a(cosα−cosβ)+b(sinα−sinβ)=0
⇒−2asin
2
α+β
sin
2
α−β
+2bsin
2
α−β
cos
2
α+β
=0
⇒2asin
2
α+β
sin
2
α−β
=2bsin
2
α−β
cos
2
α+β
⇒tan
2
α+β
=
a
b
Therefore,
sin(α+β)=
1+tan
2
2
α+β
2tan
2
α+β
=
1+
a
2
b
2
a
2b
=
a
2
+b
2
2ab
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