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Math Help - express

  1. #1
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    express

    If sin x + cos x =t , express \sin^4x+\cos^4x in terms of t .

    \sin^4x=\frac{1}{4}(1-\cos2x)^2

    =\frac{1}{4}(1-2\cos2x+\cos^22x)

    \cos^4x=\frac{1}{4}(\cos2x+1)^2

     <br />
=\frac{1}{4}(1+2\cos2x+\cos^22x)<br />

    Putting them together .

    \frac{1}{4}(1-2\cos2x+\cos^22x+1+2\cos2x+\cos^22x)

     =\frac{1}{2}(1-\cos^22x)

    I am not sure how to go on from here . . THanks for the hlep .
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  2. #2
    MHF Contributor red_dog's Avatar
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    \sin x+\cos x=t

    Square both members:

    \sin^2x+\cos^2x+2\sin x\cos x=t^2\Rightarrow 1+2\sin x\cos x=t^2\Rightarrow\sin x\cos x=\frac{t^2-1}{2}

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

    =1-2(\sin x\cos x)^2

    Now replace \sin x\cos x with \frac{t^2-1}{2}
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  3. #3
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    Quote Originally Posted by red_dog View Post
    \sin x+\cos x=t

    Square both members:

    \sin^2x+\cos^2x+2\sin x\cos x=t^2\Rightarrow 1+2\sin x\cos x=t^2\Rightarrow\sin x\cos x=\frac{t^2-1}{2}

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

    =1-2(\sin x\cos x)^2

    Now replace \sin x\cos x with \frac{t^2-1}{2}
    i complicated a simple problem .. thanks reddog .
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