If `cos^(-1)x-cos^(-1)y/2=alpha,t h e n4x^2-4x ycosalpha+y^2` is equal to (a)4 (b) `2sin^2alpha` (c) `-4sin^2alpha` (d) `4sin^2alpha`

Options

A. 4
B. `2 sin^(2) alpha`
C. `-4 sin^(2) alpha`
D. `4 sin^(2) alpha`

Explanation

We have `cos^(-1) x - cos^(-1).(y)/(2) = alpha`
or `x = cos (cos^(-1).(y)/(2) + alpha)`
`= cos (cos^(-1).(y)/(2)) cos alpha - sin (cos^(-1).(y)/(2)) sin alpha`
`= (y)/(2) cos alpha - sqrt(1 - (y^(2))/(4)) sin alpha`
or `2x = y cos alpha - sin alpha sqrt(4 -y^(2))`
or `2x - y cos alpha = - sin alpha sqrt(4 -y^(2))`
Squaring, we get
`4x^(2) + y^(2) cos^(2) alpha - 4xy cos alpha = 4 sin^(2) alpha - y^(2) sin^(2) alpha`
or `4x^(2) - 4xy cos alpha + y^(2) = 4 sin^(2) alpha`