If x = p sec θ + q tan θ and y = p tan θ + q sec θ, provethat x² - y² = p² - q².

= (p sec θ + q tan θ) ² – (p tan θ + q sec θ) ²
= p ² sec ² θ + 2pq sec θ tan θ + q ² tan ² θ - (p ² tan ² θ + 2pq tan θ sec θ + q ² sec ² θ )

= p ² sec ² θ + 2pq sec θ tan θ + q ² tan ² θ - p ² tan ² θ - 2pq tan θ sec θ -q ² sec ² θ

= (p ² - q ² ) sec ² θ + (q ² - p ² ) tan ² θ
= (p ² - q ² ) sec ² θ - (p ² - q ² ) tan ² θ

= (p ² - q ² ) (sec ² θ - tan ² θ)
= (p ² – q ² ) [Since 1 + tan ² θ = sec ² θ ]
= R.H.S.

∴ x² - y² = p ² – q ^2.
Hence, proved.