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Math Help - Trig Identity

  1. #1
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    Trig Identity

    Prove the identity

    cot\;\frac{\alpha}{2}\;=\;\frac{1 + cos\;\alpha}{sin\;\alpha}

    Any help appreciated. I'm lost here.
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  2. #2
    Super Member wingless's Avatar
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    \sin^2 \frac{\alpha}{2} = \frac{1-\cos\alpha}{2}

    \cos^2 \frac{\alpha}{2} = \frac{1+\cos\alpha}{2}


    \frac{\cos^2 \frac{\alpha}{2}}{\sin^2 \frac{\alpha}{2}} = \frac{\frac{1+\cos\alpha}{2}}{\frac{1-\cos\alpha}{2}}

    \cot^2 \frac{\alpha}{2} = \frac{1+\cos\alpha}{1-\cos\alpha}

    \cot^2 \frac{\alpha}{2} = \frac{1+\cos\alpha}{1-\cos\alpha}\cdot\frac{1+\cos\alpha}{1+\cos\alpha}

    \cot^2 \frac{\alpha}{2} = \frac{(1+\cos\alpha)^2}{1-\cos^2\alpha}

    \cot^2 \frac{\alpha}{2} = \frac{(1+\cos\alpha)^2}{\sin^2\alpha}

    \cot \frac{\alpha}{2} = \frac{1+\cos\alpha}{\sin \alpha}
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  3. #3
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    How exactly did you get that first step?
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  4. #4
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    Quote Originally Posted by Vitamin View Post
    How exactly did you get that first step?
    Half-angle identity.
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