InterviewSolution
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InterviewSolution
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If `cos^(-1) (x/2) + cos^(-1) (y/3) =alpha` then prove that `9x^2-12xycosalpha+4y^2=36sin^2alpha`. |
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Answer» `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` `=> 36(1-sin^2 alpha) - 12xy cosalpha = 36-4y^2-9x^2+x^2y^2` `=>9x^2+4y^2- 12xy cosalpha = 36sin^2alpha` |
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