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Math Help - Proving an Identity

  1. #1
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    Proving an Identity

    Prove the identity cosecA-cotA = tan(0.5A)
    Thanks in advance
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  2. #2
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by Punch View Post
    Prove the identity cosecA-cotA = tan(0.5A)
    Thanks in advance

    Try using half angle identities.

    \sin{\frac{\theta}{2}} = \sqrt{\frac{1-\cos{\theta}}{2}}

    and

    \cos{\frac{\theta}{2}} = \sqrt{\frac{1+\cos{\theta}}{2}}

    It should into an easy problem using these identities.
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  3. #3
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    Quote Originally Posted by 11rdc11 View Post
    Try using half angle identities.

    \sin{\frac{\theta}{2}} = \sqrt{\frac{1-\cos{\theta}}{2}}

    and

    \cos{\frac{\theta}{2}} = \sqrt{\frac{1+\cos{\theta}}{2}}

    It should into an easy problem using these identities.
    I haven't learnt any half angles identities or formula yet
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  4. #4
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    Have you learnt double angle identities yet?
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  5. #5
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    yes
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  6. #6
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by Punch View Post
    yes
    Ok let me show you how to derive them from the double angle

    \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}

    \cos{2\theta} = \cos^2{\theta} -(1-\cos^2{\theta})

    \cos{2\theta} = \cos^2{\theta}-1 + \cos^2{\theta}

    \cos{2\theta} = 2\cos^2{\theta} -1

    \cos{2\theta} +1= 2\cos^2{\theta}

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

    \cos{\theta} = ^+_- \sqrt{\frac{\cos{2\theta}+1}{2}}

    now replace \theta with \frac{\theta}{2}

    \cos{\frac{\theta}{2}} = ^+_- \sqrt{\frac{\cos{2(\frac{\theta}{2})}+1}{2}}

    which equals

    \cos{\frac{\theta}{2}} = ^+_- \sqrt{\frac{\cos{\theta}+1}{2}}

    Same idea to derive for sin
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  7. #7
    Super Member 11rdc11's Avatar
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    \tan{\frac{\theta}{2}} = \frac{\sin{\frac{\theta}{2}}}{\cos{\frac{\theta}{2  }}} = \frac{\sqrt{\frac{1-\cos{\theta}}{2}}}{\sqrt{\frac{1+\cos{\theta}}{2}}  } = \sqrt{\frac{1-\cos{\theta}}{1+\cos{\theta}}} = \sqrt{\frac{(1-\cos{\theta})(1-\cos{\theta})}{(1+\cos{\theta})(1-\cos{\theta)}}} = \sqrt{\frac{(1-\cos{\theta})^2}{(1-\cos^2{\theta})}}

    =  \frac{1-\cos{\theta}}{\sin{\theta}} = \frac{1}{\sin{\theta}} - \frac{\cos{\theta}}{\sin{\theta}} = \csc{\theta} - \cot{\theta}
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  8. #8
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    I don't really understand it...

    Can you show me how to start and solve the question from

    cosecA-cotA = tan(0.5A) without using half angle? sorry
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  9. #9
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    Do as you have been told.

    Use the double angle identities for sine and cosine to derive the half angle identities.

    Then do \frac{\sin{\frac{\theta}{2}}}{\cos{\frac{\theta}{2  }}}.
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  10. #10
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    Thanks guys, finally understood half angle, thankd rdc
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