If `x/a costheta+y/b sintheta=1` and `x/a sintheta-y/bcostheta=1` Prove that : `(x^2)/(a^2)+(y^2)/(b^2)=2`

Explanation

`{[frac{x}{a} cos theta+frac{y}{b} sin theta]^{2}+[frac{x}{a} sin theta-frac{y}{b} cos theta]=(1)^{2}+(1)^{2}} `

`frac{x^{2}}{a^{2}} cos ^{2} theta+frac{y^{2}}{b^{2}} sin ^{2} theta frac{2 x y}{a b} cos theta sin theta=frac{x^{2}}{a^{2}} sin ^{2} theta+frac{y^{2}}{b^{2}} cos ^{2} theta-frac{2 x y}{a b} sin theta cos theta=1+1 `

`frac{x^{2}}{a^{2}} cos ^{2} theta+frac{y^{2}}{b^{2}} cos ^{2} theta+frac{y^{2}}{b^{2}} sin ^{2} theta=2 `

`cos ^{theta}[frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}]+sin ^{2} theta(frac{x^{2}}{a^{2}}+frac{y^{2}}{a^{2}})=2 `

`frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=(therefore cos ^{2} theta+sin ^{2} theta=1)`