If `cos ^(-1)((3)/(5)), beta = tan ^(-1)((1)/(3))`, where ` -1 le x le 1, -2 le y le 2, x le (y)/(2)`, then for all `x,y, 4x^(2) - 4xy cos alpha + y^(2)` is equal to

Answer

4

Explanation

`cos^(-1) x - cos^(-1) ((y)/(2)) = alpha`
`cos^(-1) [(xy)/(2)] - sqrt(1-x^(2)) sqrt(1 - (y^(2))/(4)]= alpha`
`(xy)/(2) - sqrt(1-x^(2)) sqrt(1-(y^(2))/(4)) = cos alpha`
`(xy)/(2) - cos alpha = sqrt(1-x^(2)) sqrt(1- (y^(2))/(4))`
Squaring on both sides
`(x^(2)y^(2))/(4) + cos^(2) alpha - xy cos alpha = (1-x^(2)) (1-(y^(2))/(4))`
`cancel(x^(2)y^(2))/(4) + cos^(2) alpha - xy cos alpha = 1 - x^(2) - (y^(2))/(4) + cancel(x^(2) y^(2))/(4)`
`x^(2) + (y^(2))/(4) - xy cos alpha = sin^(2) alpha`
`4x^(2) - 4xy cos alpha + y^(2) = 4 sin^(2) alpha`