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Thread: Trig Proof

  1. #1
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    Trig Proof

    Given $\displaystyle sec^2 \theta + cosec^2 \theta = \frac{1}{sin^2\theta cos^2\theta}$

    Proof $\displaystyle sec^2 \theta + cosec^2 \theta = 4cosec^2 (2\theta).$
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Use what you've been given and use the identity:

    $\displaystyle \sin(2A) = 2\sin A\cos A$

    Can you do it? Or still stuck?
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  3. #3
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    still stuck tbh.

    does $\displaystyle cosec^2 (2\theta)$ = $\displaystyle \frac{1}{sin^2 (2 \theta)}$
    if so,
    does $\displaystyle \frac{1}{sin^2 (2\theta)} = \frac{1}{sin^2\theta cos^2\theta} + \frac{1}{ cos^2\theta sin^2\theta}$
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  4. #4
    MHF Contributor Unknown008's Avatar
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    $\displaystyle cosec^2 2\theta = \dfrac{1}{sin^2 2 \theta}$

    Ok, add the 4 now.

    $\displaystyle 4cosec^2 2\theta = \dfrac{4}{\sin^2 2 \theta}$

    Now, this can be re-written as:

    $\displaystyle 4 cosec^2 2\theta = \dfrac{4}{(\sin 2\theta)(\sin 2\theta)}$

    Use the double angle identity I gave you;

    $\displaystyle 4 cosec^2 2\theta = \dfrac{4}{(2\sin\theta\cos\theta)(2\sin\theta\cos\ theta)}$

    Can you complete it now?
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  5. #5
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    I think i can but can u just confirm for me please.

    Thanks.

    $\displaystyle 4 cosec^2 2\theta = \dfrac{4}{(2\sin\theta\cos\theta)(2\sin\theta\cos\ theta)}$

    $\displaystyle 4 cosec^2 2\theta = \frac{4}{4sin^2\theta cos^2\theta}$

    and the 4s cancel.

    $\displaystyle 4 cosec^2 2\theta = \frac{1}{sin^2\theta cos^2\theta}$

    given $\displaystyle sec^2 \theta + cosec^2 \theta = \frac{1}{sin^2\theta cos^2\theta}$

    $\displaystyle sec^2 \theta + cosec^2 \theta = 4 cosec^2 2\theta$
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  6. #6
    MHF Contributor Unknown008's Avatar
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    That's it, right!
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  7. #7
    MHF Contributor harish21's Avatar
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    (yes).
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