Given,

x=asecθ+btanθ
by squaring both side
x²=(asecθ+btanθ)²
x²=a²sec²θ+b²tan²θ+2absecθtanθ               …………….1

And,
y=atanθ+bsecθ
by squaring both side
y²=(atanθ+bsecθ)²
y²=a²tan²θ+b²sec²θ+2absecθtanθ               ………………….2

subtract eq.2 from 1
x²−y²=a²sec²θ−a²tan²θ+b²tan²θ−b²sec²θ+2absecθtanθ−2abtanθsecθ
x²−y²=a²(sec²θ−tan²θ)+b²(tan²θ−sec²θ)
x²−y² = a²(1)+b²(−1)
x²−y² = a²−b²