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

Options

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

Explanation

We have
`cos^(-1)x-cos^(-1)(y)/(2)=alpha`
`rarr cos^(-1){(xy)/(2)+sqrt(1-x^(2))sqrt(1-(y^(2))/(4))}=alpha`
`rarr(xy)/((2)+sqrt(1-x^(2))sqrt(1-(y^(2))/(4)=cos alpha`
`rarr cos^(2)alpha+_(x^(2)y^(2))/(4)-xycos alpha=1-=x^(2)-(y^(2))/(4)+(x^(2)y^(2))/(4)`
`rarr x^(2)+y^(2)/(4)-xy cos alpha= sin^(2) alpha`
`rarr 4x^(2)-4xy cos alpha+y^(2)=4 sin^(2)alpha`