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Thread: Fundamental identities

  1. #1
    Member >_<SHY_GUY>_<'s Avatar
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    Question Fundamental identities

    CAn someone explain how you get from:

    sinx cosx^2 - sinx to sinx (cos^2x-1) to -sinx (1-cos^2x) to -sinx( sin^2x) to -sin^3x

    and help me with factoring this equation:

    sec^3x-sec^2x-secx+1

    Thank you in advance
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  2. #2
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    $\displaystyle {\color{blue}\sin x} \cos^{2} x - {\color{blue}\sin x}$
    $\displaystyle = {\color{blue}\sin x} \left(\cos^{2} x - 1\right)$

    sin x was factored out of the expression just as you would if you had something like $\displaystyle ab + a = a(b + 1)$

    $\displaystyle = -\sin x \left(1 - cos^{2} x\right)$ (-1 was factored out of the bracketed expression)

    Recall that: $\displaystyle \cos^{2} x + \sin^{2} x = 1 \: \Rightarrow \: {\color{red}\sin^{2}x = 1 - \cos^{2} x}$

    Substituting it in, we get:
    $\displaystyle = -\sin x \cdot {\color{red} \sin^{2} x}$
    $\displaystyle = -\sin^{3} x$ since $\displaystyle a^{m}a^{n} = a^{m+n}$

    ----------------------------
    $\displaystyle {\color{red}\sec^{3} x - \sec^{2} x} \: {\color{blue}- \sec x + 1}$

    Focusing on the red, let's factor out $\displaystyle \sec^{2} x$ and for the blue, let's factor out -1:
    $\displaystyle = \sec^{2} x (\sec x - 1) -( \sec x - 1)$

    Factor out (sec x - 1):
    $\displaystyle = (\sec x - 1){\color{magenta}\left(sec^{2}x - 1\right)}$

    And hopefully the expression in purple reminds you of a certain identity.
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  3. #3
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    so from that i use the pythagorean identities multiply, and thats it right?
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    I'm not sure what you mean by what you wrote. What I was referring to was this identity: $\displaystyle 1 + \tan^{2} x = \sec^{2} x$
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    which is derived from the pythagorean identity: sin^2+cos^2 = 1
    and from there you multiply
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