InterviewSolution
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InterviewSolution
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If `cos ^(-1) .(x)/(a)+ cos ^(-1). (y)/(b) = theta` then prove that `(x^(2))/(a^(2)) -(2xy)/(ab) . cos theta+ (y^(2))/(b^(2)) = sin ^(2) theta`. |
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Answer» `cos^(-1).(x)/(a) + cos theta +(y^(2))/(b^(2)) = sin^(2) theta` `rArr cos^(-1)[(x)/(a).(y)/(b)-sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))] = theta` `rArr (xy)/(ab) -sqrt(1-(x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(2)y^(2))/(a^(2)b^(2)))= cos theta` `rArr " " (xy)/(ab)-costheta =sqrt(1-(x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(3)y^(2))/(a^(2)b^(2)))` squaring both sideswe get `(x^(2)y^(2))/(a^(2)b^(2))+ cos^(2) theta -(2xy)/(ab) cos theta = 1- (x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(2)y^(2))/(a^(2)b^(2))` `rArr (x^(2))/(a^(2)) -(2xy)/(ab) cos theta +(y^(2))/(b^(2)) = 1cos^(2) theta` `rArr (x^(2))/(y^(2)) -(2xy)/(ab) costheta +(y^(2))/(b^(2)) = sin^(2)theta` Hence Proved. |
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