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Prove the following identities(secx sec y + tanx tan y)2 – (secx tan y + tanx sec y)2 = 1 |
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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 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. |
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