Prove that tan^2 x + cot^2 x = sec^2 x + cosec^2 x - 2
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Prove that tan^2 x + cot^2 x = sec^2 x + cosec^2 x - 2

[From: ] [author: ] [Date: 11-11-07] [Hit: ]
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help me..tq vm..

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tan^2 x + cot^2 x

= sin^2 x / cos^2 x + cos^2 x / sin^2 x

= (sin^4 x + cos^4 x) / sin^2 x cos^2 x

= ((1 - cos^2 x)^2 + cos^4 x) / sin^2 x cos^2 x

= (1 - 2 cos^2 x + cos^4 x + cos^4 x) / sin^2 x cos^2 x

= (1 - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x ; (use 1 = sin^2 x + cos^2 x)

= (sin^2 x + cos^2 x - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x

= (sin^2 x - cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x

= 1/cos^2 x - 1/sin^2 x + 2 cos^2 x / sin^2 x ; ( cos^2 x = 1 - sin^2 x)

= 1/cos^2 x - 1/sin^2 x + 2 (1 - sin^2 x)/ sin^2 x

= 1/cos^2 x - 1/sin^2 x + 2 /sin^2 x - 2 sin^2 x/ sin^2 x

= 1/cos^2 x + 1/sin^2 x - 2

= sec^2 x + csc^2 x - 2
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