LHS = (secx sec y + tanx tan y)² – (secx tan y + tanx sec y)²

= [(secx sec y)² + (tanx tan y)² + 2 (secx sec y) (tanx tan y)] – [(secx tan y)² + (tanx sec y)² + 2 (secx tan y) (tanx sec y)]

= [sec²x sec² y + tan²x tan² y + 2 (secx sec y) (tanx tan y)] – [sec²x tan² y + tan²x sec² y + 2 (sec²x tan² y) (tanx sec y)]

= sec²x sec² y - sec²x tan² y + tan²x tan² y - tan²x sec² y

= sec²x (sec² y - tan² y) + tan²x (tan² y - sec² y)

= sec²x (sec² y - tan² y) - tan²x (sec² y - tan² y)

We know that sec²x – tan²x = 1.

= sec²x × 1 – tan²x × 1

= sec²x – tan²x

= 1

= RHS

Hence proved.