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Thread: Verifying Identities!

  1. #1
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    Verifying Identities!

    I need help on these:

    1+CSC
    COT+COS = SEC



    SIN + COS = 1
    CSC SEC


    1+ CSC - COT= COS
    SEC


    Theres theta symbols after the identities. I just did'nt know how to put them in. haha.
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  2. #2
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    Hello, Lauren!

    $\displaystyle \frac{1+\csc\theta}{\cot\theta + \cos\theta} \:=\:\sec\theta$

    The left side is: .$\displaystyle \frac{1 + \dfrac{1}{\sin\theta}}{\dfrac{\cos\theta}{\sin\the ta} + \cos\theta} $

    Multiply by $\displaystyle \tfrac{\sin\theta}{\sin\theta}\!:\quad\frac{\sin\t heta}{\sin\theta}\cdot\frac{1 + \dfrac{1}{\sin\theta}}{\dfrac{\cos\theta}{\sin\the ta} + \cos\theta} \;=\;\frac{\sin\theta + 1}{\cos\theta + \sin\theta\cos\theta} $


    Factor and reduce: .$\displaystyle \frac{\sin\theta + 1}{\cos\theta(1 + \sin\theta)} \;=\;\frac{1}{\cos\theta} \;=\; \sec\theta $




    $\displaystyle \frac{\sin\theta}{\csc\theta} + \frac{\cos\theta}{\sec\theta} \:=\:1$

    We have: .$\displaystyle \frac{\sin\theta}{\frac{1}{\sin\theta}} + \frac{\cos\theta}{\frac{1}{\cos\theta}} \;=\;\sin^2\!\theta + \cos^2\!\theta \;=\;1$




    $\displaystyle \frac{1 + \csc\theta}{\sec\theta} - \cot\theta \:=\:\cos\theta$
    We have: .$\displaystyle \frac{1 + \dfrac{1}{\sin\theta}}{\dfrac{1}{\cos\theta}}- \frac{\cos\theta}{\sin\theta}$

    Multiply by $\displaystyle \tfrac{\sin\theta\cos\theta}{\sin\theta\cos\theta} \!:\quad
    \frac{\sin\theta\cos\theta}{\sin\theta\cos\theta}\ cdot\frac{1 + \dfrac{1}{\sin\theta}}{\dfrac{1}{\cos\theta}} - \frac{\cos\theta}{\sin\theta} \;=\;\frac{\sin\theta\cos\theta + \cos\theta}{\sin\theta} - \frac{\cos\theta}{\sin\theta}$


    . . $\displaystyle = \;\frac{\sin\theta\cos\theta + \cos\theta - \cos\theta}{\sin\theta} \;=\;\frac{\sin\theta\cos\theta}{\sin\theta} \;=\;\cos\theta $

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