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Math Help - establishing identities?

  1. #1
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    Exclamation establishing identities?

    how to prove?

    (cot^2 x-1) / (2 cot x) = cot 2x

    cos 3x = 4 cos^3 x-3 cosx

    sin 2t+sin t/1+cos t+sos 2t = tan t

    THANK YOU!
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  2. #2
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    \frac{cot^2x-1}{2cotx}

    =\frac{\frac{1-tan^2x}{tan^2x}}{\frac{2}{tanx}}

    =\frac{1-tan^2x}{2tanx}

    =cot2x

    Knowing the fact that cot x = 1/tan x .
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  3. #3
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    Quote Originally Posted by pongpong View Post
    how to prove?

    cos 3x = 4 cos^3 x-3 cosx
    cos(2x+x)=cos2xcosx-sin2xsinx

    =(2cos^2x-1)cosx-2sinxcosx(sinx)

    =2cos^3x-cosx-2sin^2xcosx

    =2cos^3x-cosx-2(1-cos^2x)(cosx)

    =2cos^3x-cosx-2cosx+2cos^3x

    =4cos^3x-3cosx
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  4. #4
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    Hello, pongpong!

    The third one is impossible to read.

    Well, not "impossible" . . . I figured it out!

    \frac{\sin2t+\sin t}{1+\cos t+\cos 2t} \:=\: \tan t

    We're expected to know these Double-Angle Identities:

    . . . . . \begin{array}{ccc}\sin2\theta &=& 2\sin\theta\cos\theta \\ \cos2\theta &=& 2\cos^2\!\theta - 1\end{array}


    The left side becomes: . \frac{2\sin t\cos t + \sin t}{1 + \cos 2 + 2\cos^2\!t - 1} \;=\;\frac{2\sin t\cos t + \sin t}{2\cos^2\!t + \cos t}


    Factor and reduce: . \frac{\sin t(2\cos t + 1)}{\cos t(2\cos t - 1)} \;=\;\frac{\sin t}{\cos t} \;=\;\tan t

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