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Math Help - Proving identities..

  1. #1
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    Proving identities..

    Hi ya'll.

    Another question that has me completely and utterly stumped.


    show that:

    ((1+cot theta)/csc theta)) - (sec theta/(tan theta + cot theta))= cos theta


    sorry about not being able to write it properly.
    If you could explain what you did as well so i can follow on, that would be amazing.

    Thanks
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  2. #2
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    Re: Proving identities..

    Quote Originally Posted by FlexedCookie View Post
    Hi ya'll.

    Another question that has me completely and utterly stumped.


    show that:

    ((1+cot theta)/csc theta)) - (sec theta/(tan theta + cot theta))= cos theta


    sorry about not being able to write it properly.
    If you could explain what you did as well so i can follow on, that would be amazing.

    Thanks
    (\sin \theta)(1+\cot \theta)-\frac{1}{(\cos \theta) (\tan \theta + \cot \theta)}=\cos \theta

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

    \sin \theta + \cos \theta-\frac{\sin \theta}{1}=\cos \theta

     \cos \theta = \cos \theta
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  3. #3
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    Re: Proving identities..

    Hello, FlexedCookie!

    \text{Show that: }\:\dfrac{1+\cot\theta}{\csc\theta} - \dfrac{\sec\theta}{\tan\theta + \cot\theta} \:=\:\cos \theta

    \text{We have: }\:\dfrac{1 + \frac{\cos\theta}{\sin\theta}}{\frac{1}{\sin\theta  }} - \dfrac{\frac{1}{\cos\theta}}{\frac{\sin\theta}{ \cos\theta} + \frac{\cos\theta}{\sin\theta}}


    Multiply the first fraction by \tfrac{\sin\theta}{\sin\theta}, the second fraction by \tfrac{\sin\theta\cos\theta}{\sin\theta\cos\theta}

    . . \displaystyle \frac{\sin\theta\left(1 + \frac{\cos\theta}{\sin\theta}\right)}{\sin\theta \left(\frac{1}{\sin\theta}\right)} - \frac{\sin\theta\cos\theta\left(\frac{1}{\cos\thet  a}\right)}{\sin\theta\cos\theta\left(\frac{ \sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}\right)}

    . . \displaystyle=\;\frac{\sin\theta + \cos\theta}{1} - \frac{\sin\theta}{\underbrace{\sin^2\theta + \cos^2\theta}_{\text{This is 1}}}

    . . =\;\sin\theta + \cos\theta - \sin\theta

    . . =\;\cos\theta
    Thanks from FlexedCookie
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  4. #4
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    Re: Proving identities..

    Thank you do much! I think i was losing it when there was just too many sin and cos. got me slightly confused.
    Thanks!
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