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 tan2y + tan2x tan2y - tan2x sec2 y
= sec2x (sec2y - tan2y) + tan2x (tan2y - sec2y)
= sec2x (sec2y - tan2y) - tan2x (sec2y - tan2y)
We know that sec2x – tan2x = 1.
= sec2x × 1 – tan2x × 1
sec2x – tan2x
= 1
= RHS
Hence proved.
