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Math Help - terms of powers (complete numbers)!

  1. #1
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    terms of powers (complete numbers)!

    express \cos^4\theta in terms of powers of \tan\theta and \cos\theta.


    pls help on the above problem, thanks
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  2. #2
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    Re: terms of powers (complete numbers)!

    I'll write x instead of theta
    cos^4x=(1-sin^2x)^2=1-2sin^2x+sin^4x
    But sinx=tanxcosx
    So cos^4x=1-2tan^2xcos^2x+tan^4xcos^4x
    Thanks from lawochekel
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  3. #3
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    Re: terms of powers (complete numbers)!

    Quote Originally Posted by lawochekel View Post
    express \cos^4\theta in terms of powers of \tan\theta and \cos\theta.


    pls help on the above problem, thanks
    \tan^2 \theta = \frac{1-\cos^2 \theta}{\cos^2 \theta} \Rightarrow \cos^2 \theta = \frac{1}{1+\tan^2 \theta}

    Hence :

    \cos^4 \theta =\frac{\cos^2 \theta}{1+\tan^2 \theta}
    Thanks from lawochekel
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  4. #4
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    Re: terms of powers (complete numbers)!

    Hello, lawochekel!

    A slightly different approach . . .


    \text{Express }\cos^4\theta\text{ in terms of powers of }\tan\theta\text{ and }\cos\theta.

    \cos^4\!\theta \;=\;\cos^2\!\theta\cdot\cos^2\!\theta \;=\;\frac{\cos^2\!\theta}{\sec^2\!\theta} \;=\;\frac{\cos^2\!\theta}{\tan^2\!\theta + 1}
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  5. #5
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    Re: terms of powers (complete numbers)!

    Or an even easier approach cos^4(theta) = cos^4(theta) + tan^0(theta) - 1
    Thanks from lawochekel
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