Question

If $x=asec\theta +btan\theta$ and $y=atan\theta +bsec\theta$, prove that

$(x^2-y^2)=(a^2-b^2)$.

Answer 1

$x^2-y^2$

$=(asec\theta +btan\theta {)}^2-(atan\theta +bsec\theta {)}^2$

$=(a^{2}se{c}^2\theta +b^{2}ta{n}^2\theta +2absec\theta tan\theta )-(a^{2}ta{n}^2\theta +b^{2}se{c}^2\theta +2abtan\theta sec\theta )$

$=a^2(se{c}^2\theta -ta{n}^2\theta )+b^2(ta{n}^2\theta -se{c}^2\theta )$

$=a^2-b^2$