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Math Help - Prove tan^4 + tan^2 + 1 = 1 - Sin^2 Cos^2

  1. #1
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    Prove tan^4 + tan^2 + 1 = 1 - Sin^2 Cos^2

    tan^4 x + tan^2 x+ 1 = 1 - (Sin^2 x)(Cos^2 x)

    Please someone help me solve this one so I can get some sleep tonight! I've been trying for 3 days off and on. Pretty please!

    Carol
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  2. #2
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    This identity is false. Are you sure you copied it correctly?
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  3. #3
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    tan^4Ө + tan^2Ө + 1 = 1 + (sin^2Ө)(cos^2Ө)

    Here it is, pasted directly from the homework. If it's false, I will be relieved. Is there a way to prove that it's false?

    Carol
    Last edited by CarolChang; December 11th 2008 at 05:14 PM. Reason: Edited because it needed some ^'s
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  4. #4
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    Working on the left hand side...

    tan^2x + tan^4x +1
    = \frac{1}{cos^2x} + tan^4x ... the entire left hand side can be manipulated using this.. 1 +tan^2x= sec^2 x
    = \frac{1}{cos^2x} + \left(\frac{1}{cos^2x} -1 \right) \left(\frac{1}{cos^2x} -1 \right)
    = \frac{1}{cos^2x} + \left(\frac{1}{cos^2x} -1 \right) \left(\frac{1}{cos^2x} -1 \right)
    = \frac{1}{cos^4x} - \frac{1}{cos^2x} +1
     = \frac{1}{cos^4x} - tan^2x
     = 1

    Is the RHS = 1?
    Last edited by Scopur; December 14th 2008 at 09:12 PM.
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