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

y=`cos^(-1)(x/a)+cos^(-1)(y/b)=alpha,`

=`cos^(-1)[x/a​⋅(y/b)​−sqrt((1−x^2/a^2​)(1−y^2/b^2​))
​]=α`

`cos(α)=(xy)/(ab)-sqrt((1−x^2/a^2​)(1−y^2/b^2)​)`

=`sqrt((1−x^2/a^2​)(1−y^2/b^2​))=(xy)/(ab)-cos(α)`

`(1−x^2/a^2​)(1−y^2/b^2​)=((xy)/(ab))^2((xy)/(ab))cos(α)+cos(α)^2`

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