Answer :

We are given with,


We need to find the value of 9x2 – 12xy cos θ + 4y2.


Using property of inverse trigonometry,



Take Left Hand Side (LHS) of:



Replace A by and B by .






Further solving,



We shall equate LHS to RHS,



Taking cosine on both sides,



Using property of inverse trigonometry,


cos(cos-1 A) = A


So,





By cross-multiplying,


xy - √(4 – x2) √(9 – y2) = 6 cos θ


Rearranging it,


xy – 6 cos θ = √(4 – x2) √(9 – y2)


Squaring on both sides,


[xy – 6 cos θ]2 = [√(4 – x2) √(9 – y2)]2


Using algebraic identity,


(a – b)2 = a2 + b2 – 2ab


(xy)2 + (6 cos θ)2 – 2(xy)(6 cos θ) = (4 – x2)(9 – y2)


x2y2 + 36 cos2 θ – 12xy cos θ = 36 – 9x2 – 4y2 + x2y2


x2y2 – x2y2 + 9x2 – 12xy cos θ + 4y2 = 36 – 36 cos2 θ


9x2 – 12xy cos θ + 4y2 = 36 (1 – cos2 θ)


Using trigonometric identity,


sin2 θ + cos2 θ = 1


sin2 θ = 1 – cos2 θ


Substituting the value of (1 – cos2 θ), we get


9x2 – 12xy cos θ + 4y2 = 36 sin2 θ

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