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Thread: Trigonometric Identities

  1. #1
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    Trigonometric Identities

    Hello all, I have done a question but I just need clarification if it is correct:

    Cot (x/2) = (1 + cos(x)) / sin(x)

    what is the largest set of values of x for which this identity is valid?

    i reasoned here that because they are equal to one another that either function can be used to solve for a value. this meant that for the identity to be invalid both functions would have to have a sin value of 0. this only happens at 2 * n * pi where n is any integer.

    Am I correct?
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  2. #2
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    Re: Trigonometric Identities

    Well if the equation doesn't hold true for all values of $\displaystyle \displaystyle x$, then it's not an identity.

    $\displaystyle \displaystyle \begin{align*} \frac{1 + \cos{x}}{\sin{x}} &= \frac{1 + \cos{\left(2\cdot \frac{x}{2}\right)}}{\sin{\left(2\cdot \frac{x}{2}\right)}} \\ &= \frac{1 + \cos^2{\frac{x}{2}} - \sin^2{\frac{x}{2}}}{2\sin{\frac{x}{2}}\cos{\frac{ x}{2}}} \\ &= \frac{2\cos^2{\frac{x}{2}}}{2\sin{\frac{x}{2}}\cos {\frac{x}{2}}} \\ &= \frac{\cos{\frac{x}{2}}}{\sin{\frac{x}{2}}} \\ &= \cot{\frac{x}{2}}\end{align*}$

    It's true for all $\displaystyle \displaystyle x$.
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    Re: Trigonometric Identities

    gah, I suck

    Just a quick question though, since cot (x/2) = cos (x/2) / sin (x/2) how come you didnt use sin (x/2) as a restriction since it cannot ever be equal to zero?\

    thanks in advance
    Last edited by andrew2322; Aug 6th 2011 at 10:49 PM.
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  4. #4
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    Re: Trigonometric Identities

    Quote Originally Posted by andrew2322 View Post
    gah, I suck

    Just a quick question though, since cot (x/2) = cos (x/2) / sin (x/2) how come you didnt use sin (x/2) as a restriction since it cannot ever be equal to zero?\

    thanks in advance
    This explains:
    Quote Originally Posted by Prove It View Post
    Well if the equation doesn't hold true for all values of $\displaystyle \displaystyle x$, then it's not an identity.
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Trigonometric Identities

    @mathgirl8:

    You have to use [ tex ] ... [ /tex ], not [ tex ] ... [ \tex ]
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