Prove that(a very interesting one)

Posted June 20th, 2008 by alpha

If x/a cos theta+y/b sin theta=1
and x/a sin theta-y/b cos theta=1

Then prove that x^2/a^2+y^2/b^2=2

Show all the steps please.

Comments

On June 20th, 2008 Structure says:

$\frac{x}{a}\cos \theta+\frac{y}{b}\sin \theta=1$

$\frac{x}{a}\sin \theta-\frac{y}{b}\cos \theta=1$

then

$(\frac{x}{a}\cos \theta+\frac{y}{b}\sin \theta)^2=\frac{x^2}{a^2}\cos^2 \theta+2\frac{xy}{ab}\sin\theta\cos \theta+\frac{y^2}{b^2}\sin^2\theta=1$

$(\frac{x}{a}\sin\theta-\frac{y}{b}\cos \theta)^2=\frac{x^2}{a^2}\sin^2 \theta-2\frac{xy}{ab}\sin\theta\cos \theta+\frac{y^2}{b^2}\cos^2\theta=1$

Summing up these two equalities you have

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=2$