Question

if x = p sec theta + q tan theta and y = p tan theta + q sec theta then prove that x square - y square = p square - q square

Answer 1

X=psecθ+qtanθ and y=ptanθ+qsecθ
∴, x²=p²sec²θ+2pqsecθtanθ+q²tan²θ and
y²=p²tan²θ+2pqtanθsecθ+q²sec²θ
∴, x²-y²
=p²sec²θ+2pqsecθtanθ+q²tan²θ-p²tan²θ-2pqsecθtanθ-q²sec²θ
=sec²θ(p²-q²)-tan²θ(p²-q²)
=(p²-q²)(sec²θ-tan²θ)
=(p²-q²).(1) [∵, sec²θ-tan²θ=1]
=p²-q² (Proved)