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Math Help - [Trig Proof] Does this answer key make sense?

  1. #1
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    [Trig Proof] Does this answer key make sense?

    Hi,

    Does the answer key section here for number 2 makes sense? It seems like they went from cos^2 - sin^2 to cos^2 + sin^2 for no reason.

    http://i.imgur.com/bdyQPrJ.png

    Thanks.
    Last edited by KevinShaughnessy; June 29th 2013 at 03:53 PM.
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  2. #2
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    Re: [Trig Proof] Does this answer key make sense?

    Hello, KevinShaughnessy!

    What they did was correct.


    \text{Prove: }\:\frac{\sin2t}{\sin t} - \frac{\cos2t}{\cos t} \:=\:\sec t


    \frac{\sin2t}{\sin t} - \frac{\cos2t}{\cos t} \;=\;\frac{2\sin t\cos t}{\sin t} - \frac{(\cos^2\!t-\sin^2\!t)}{\cos t}

    n . . . . . . . . . =\;2\cos t - \left(\frac{\cos^2\!t-\sin^2\!t}{\cos t}\right)

    n . . . . . . . . . =\;\frac{2\cos^2\!t - (\cos^2\!t - \sin^2\!t)}{\cos t}

    n . . . . . . . . . =\;\frac{2\cos^2\!t - \cos^2\!t + \sin^2t}{\cos t}

    n . . . . . . . . . =\;\frac{\cos^2\!t + \sin^2\!t}{\cos t}

    n . . . . . . . . . =\;\frac{1}{\cos t}


    n . . . . . . . . . =\;\sec t
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  3. #3
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    Re: [Trig Proof] Does this answer key make sense?

    Yes, the answer key is perfectly correct. The problem is to prove the trig identity \frac{sin(2t)}{sin(t)}- \frac{cos(2t)}{cos(t)}= sec(t). They do that by using the "double angle" identities: sin(2t)= 2 sin(t)cos(t) and cos(2t)= cos^2(t)- sin^2(t) to write the left side as \frac{2sin(t)cos(t)}{sin(t)}- \frac{cos^2(t)- sin^2(t)}{cos(t)}.

    Of course we can cancel the "sin(t)" terms in the first fraction to get
    2 cos(t)- \frac{cos^2(t)- sin^2(t)}{cos(t)}
    and turn the first "2cos(t)" into a fraction by multiplying both numerator and denominator by cos(t):
    \frac{2cos^2(t)}{cos(t)}- \frac{cos^2(t)- sin^2(t)}{cos(t)}.

    Now add the fractions, remembering to distribute that "-" in front of the second fraction:
    \frac{2cos^2(t)- cos^2(t)+ sin^2(t)}{cos(t)}= \frac{cos^2(t)+ sin^2(t)}{cos(t)}= \frac{1}{cos(t)}= sec(t).
    Last edited by HallsofIvy; June 29th 2013 at 04:52 PM.
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  4. #4
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    Re: [Trig Proof] Does this answer key make sense?

    Thanks guys, looks like I need a bit of work on knowing when to distribute a negative.
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