If ^-1 x - ^-1 y/2 = α , then 4x^2 - 4xy α + y^2 is equal to: - Mathematics MCQ AIEEE 2026 | ExamTest.live

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If cos^{- 1} x - cos^{- 1} \frac{y}{2} = α, then 4 x^{2} - 4 x y cos α + y^{2} is equal to:

Solution & Step-by-step Explanation

Using the identity cos^{- 1} A - cos^{- 1} B = cos^{- 1} [A B + (1 - A^{2}) (1 - B^{2})] : cos^{- 1} [x \cdot \frac{y}{2} + (1 - x^{2}) (1 - y^{2} /4)] = α \frac{x y}{2} + (1 - x^{2}) (1 - y^{2} /4) = cos α (1 - x^{2}) (1 - y^{2} /4) = cos α - \frac{x y}{2} Squaring both sides: (1 - x^{2}) (1 - y^{2} /4) = cos^{2} α + \frac{x ^{2} y ^{2}}{4} - x y cos α 1 - \frac{y ^{2}}{4} - x^{2} + \frac{x ^{2} y ^{2}}{4} = cos^{2} α + \frac{x ^{2} y ^{2}}{4} - x y cos α 1 - \frac{y ^{2}}{4} - x^{2} = cos^{2} α - x y cos α Multiplying by 4: 4 - y^{2} - 4 x^{2} = 4 cos^{2} α - 4 x y cos α 4 x^{2} - 4 x y cos α + y^{2} = 4 - 4 cos^{2} α 4 x^{2} - 4 x y cos α + y^{2} = 4 (1 - cos^{2} α) = 4 sin^{2} α

Try it yourself before checking the explanation above.

If cos^{- 1} x - cos^{- 1} \frac{y}{2} = α, then 4 x^{2} - 4 x y cos α + y^{2} is equal to:

2 sin 2 α

4

4 sin^{2} α

- 4 sin^{2} α

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