Question

If x= psecθ + qtanθ and y = ptanθ + q secθ, then prove that x 2 – y 2 = p 2 –q 2

Answer 1

x²= (psecθ + qtanθ)

p²sec²θ+q²tan²θ+2psecθtanθ

y² = (ptanθ + q secθ)²

q²sec²θ+p²tan²θ+2psecθqtanθ

x²-y²=p²sec²θ+q²tan²θ+2psecθqtanθ-q²sec²θ-p²tan²θ-2psecθqtanθ

x²-y²=p²sec²θ-p²tan²θ-q²sec²θ+q² tan²θ

x²-y²=p²(sec²θ-tan²θ) - q²(sec²θ-tan²θ)

x²-y²=(sec²θ-tan²θ) (p²-q²)

sec²θ-tan²θ = 1

x²-y²=p²-q²