Question

if x=p secQ+qtanQ and y=ptanQ+qsecQ
prove that x^2-y^2=p^2-q^2​

Answer 1

x = p secθ + q tanθ and y = p tanθ + q secθ

L.H.S. = x2 - y2

= (p secθ + q tanθ)2 - (p tanθ + q secθ)2

= p2 sec2θ + 2pq secθ tanθ + q2 tan2θ - (p2tan2θ + 2pq tanθ secθ + q2sec2θ)

= p2sec2θ + 2pq secθ tanθ + q2 tan2θ - p2 tan2θ - 2pq tanθ secθ - q2 sec2θ

= (p2-q2) sec2θ + (q2-p2) tan2θ

= (p2-q2) sec2θ + (q2-p2) tan2θ = (p2-q2) (sec2θ - tan2θ)

= (p2-q2) [since 1 + tan2θ = sec2θ]

= R.H.S. ∴ x2-y2 = p2-q2.