If `cos^(-1)x/a+cos^(-1)y/b=alpha,` prove that `(x^2)/(a^2)-2(x y)/(a b)cosalpha+(y^2)/(b^2)=sin^2alpha`

Explanation

`cos^(-1)[x/a. y/b-sqrt((1-frac{x^2}{a^2})(1-frac{y^2}{b^2}))]=alpha`

`cos alpha= frac{xy}{ab}-sqrt ((frac{x^2}{a^2})(1-frac{y^2}{b^2}))`

`sqrt((frac{x^2}{a^2})(1-frac{y^2}{b^2}))= frac{xy}{ab}-cos alpha`

`(frac{x^2}{a^2})(1-frac{y^2}{b^2})=(frac{xy}{ab})^2-frac{2xy}{ab}cos alpha+cos^2 alpha`

`1-cos^2 alpha=frac{y^2}{b^2}-frac{2xy}{ab}cos alpha+frac{x^2}{a^2}`

`(x^2)/(a^2)-2(x y)/(a b)cosalpha+(y^2)/(b^2)=sin^2alpha`