Question

If cos-1 x - cos-1 y/2 = ​alpha, then 4x2 - 4xy cos alpha + y2 = ? (a) 4 (b) 2sin2 alpha (c) -4sin2 alpha (d) 4sin2 alpha

Answer 1

Please see the attachment

Answer 2

given, cos-¹x - cos-¹(y/2) = α

we have to find value of 4x² - 4xycosα + y²

cos-¹x - cos-¹(y/2) = α

⇒cos-¹{xy/2 - √(1 - x²)√(2² - y²)/2} = α

⇒cosα = xy/2 - √(1 - x²)(4 - y²)/2

⇒2cosα = xy - √(4 - 4x² - y² + x²y²)

⇒(xy - 2cosα)² = √(4 - 4x² - y² + x²y²)

squaring both sides we get,

⇒x²y² + 4cos²α - 4xycosα = 4 - 4x² - y² + x²y²

⇒4x² - 4xycosα + y² = 4 - 4cos²α = 4(1 - cos²α)

⇒4x² - 4xycosα + y² = 4sin²α

hence it is clear that, 4x² - 4xycosα + y² = 4sin²α

option (d) is correct choice.

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