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If cos^(-1) (x/2) + cos^(-1) (y/3) =alph...

If `cos^(-1) (x/2) + cos^(-1) (y/3) =alpha` then prove that `9x^2-12xycosalpha+4y^2=36sin^2alpha`.

Text Solution

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`cos^-1(x/2)+cos^-1(y/3) = alpha`
`=>cos^-1((x/2)(y/3) - sqrt(1-(x^2)/4)sqrt(1-y^2/9)) = alpha`
`=>((xy)/6) - sqrt((4-x^2)/4)sqrt(9-y^2)/9) = cosalpha`
`=>xy - sqrt(4-x^2)sqrt(9-y^2) = 6cos alpha`
`=>xy - 6cos alpha= sqrt(4-x^2)sqrt(9-y^2)`
Squaring both sides,
`=>x^2y^2+36 cos^2 alpha - 12xy cosalpha = (4-x^2)(9-y^2)`
`=>x^2y^2+36 cos^2 alpha - 12xy cosalpha = 36-4y^2-9x^2+x^2y^2`
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