If `cos^(-1)x/2+cos^(-1)y/3=alpha` , then prove that `9x^2-12 x ycosalpha+4y^2=36\ s in^2alpha` .

Options

A. 36
B. `-36 sin^(2) theta`
C. `36 sin^(2) theta`
D. `36 cos^(2) theta`

Explanation

We have
`cos^(-1)(x)/(2)+cos^(-1)(y)/(3)=theta`
`rarr cos^(-1)(xy)/(6)sqrt(1-(x^(2))/(4)sqrt(1-(y^(2))/(9))}=theta`
`rarr xy-sqrt(4-x^(2))sqrt(9-y^(2))=6costheta`
`rarr xy-6cos theta^(2)=(4-x^(2))(9-y^(2))`
`rarr-12xy cos theta + 36 cos^(2) theta =36 -4y^(2)-9x^(2)`
`9x^(2)+4y^(2)-12xy cos theta =36 sin^(2) theta`