Answer :

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.


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