## Book: RD Sharma - Mathematics (Volume 1)

### Chapter: 4. Inverse Trigonometric Functions

#### Subject: Maths - Class 12th

##### Q. No. 17 of MCQ

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17
##### Choose the correct answerIf , then 9x2 – 12xy cos θ + 4y2 is equal to

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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