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Verifying Trig. Identities

  1. Jan 12, 2012 #1
    Any/All help is appreciated :) Thanks!

    1. The problem statement, all variables and given/known data

    All that has to be done is proving that these two sides are equal. Basically, you just work through the problem until both sides are the same.

    (csc(x)-sec(x))/(csc(x)+sec(x)) = (tan(x)-1)/(tan(x)+1)



    2. Relevant equations

    sin2x + cos2x = 1

    csc(x) = 1/sin(x)

    sec(x) = 1/cos(x)

    tan(x) = sin(x)/cos(x)

    sin(-x) = -sin(x)

    cos(-x) = -sin(x)



    3. The attempt at a solution

    coverted to terms of sin and cos
    (1/sinx-1/cosx)/(1/sinx+1/cosx) = ((sinx/cosx)-1)/((sinx/cosx)+1)

    flipped and multiplied, then started simplifying
    sinx/sinx - sinx/sinx + cosx/sinx - sinx/cosx = sinxcosx/sinxcosx + sinx/cosx - cosx/sinx - 1

    continued simplifying
    cosx/sinx - sinx/cosx = 1 - 1 + sinx/cosx -cosx/sinx

    cosx/sinx - sinx/cosx = sinx/cosx - cosx/sinx

    This is where I got confused. I'm not sure how to get the sides equal now. I tried a few things...not sure if they're right... I don't know how to make a -cos2x into cos2x and same with the sin.

    multiplied in order to get common denominators

    (cos2x-sin2x)/sinxcosx = (sin2x-cos2x)/sinxcosx
     
  2. jcsd
  3. Jan 12, 2012 #2

    jedishrfu

    Staff: Mentor

    I think you have to pick one side or the other and thru transformation derive the other side. So I'd start with the tan side first, convert to sin and cos then notice the numerator can be factored from (s/c - 1) to (s/c - c/c) to (s - c) /c and similarly for the denominator and you're almost home.
     
  4. Jan 12, 2012 #3

    SammyS

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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Hello Aubrie. Welcome to PF !

    There's a typo or other mistake in one of you 'equations' above.

    When you flipped and multiplied, did you 'flip' (1/sinx+1/cosx) and get sin(x)+cos(x). If you did then that's a BIG algebra no-no .

    [itex]\displaystyle\frac{1}{\displaystyle\frac{1}{\sin(x)}+\frac{1}{\cos(x)} }\ne \sin(x)+\cos(x)[/itex]

    After you get (1/sin(x)-1/cos(x))/(1/sin(x)+1/cos(x)) on the LHS, multiply the numerator & denominator by sin(x):
    [itex]\displaystyle\frac{\displaystyle\sin(x)\left(\frac{1}{sin(x)}-\frac{1}{\cos(x)}\right)}{\displaystyle\sin(x) \left( \frac{1}{sin(x)}+\frac{1}{\cos(x)}\right)}[/itex]​

    See where that takes you.
     
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