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Math Help - [SOLVED] Proving trig identities

  1. #1
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    Post [SOLVED] Proving trig identities

    Prove the following trig identities starting with the left side.

    1) sin^2x + cos^2x = 1

    2) (cosx - sinx)^2 + (cosx + sinx)^2 = 2

    3) cotx + tanx = secx*cscx

    Thank you for any help!
    Last edited by live_laugh_luv27; April 28th 2009 at 02:52 PM.
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  2. #2
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    Quote Originally Posted by live_laugh_luv27 View Post
    Prove the following trig identities starting with the left side.

    1) sin^2x + cos^2x = 1

    2) (cosx - sinx)^2 + (cosx + sinx)^2 = 2

    3) cotx + tanx = secx*cscx

    Thank you for any help!
    1. This seems very weird, it's pretty much the basic proof.

    sin(x) = opp/hyp
    cos(x) = adj/hyp

    Therefore if we square both:

    sin^2(x) = (opp)^2/(hyp)^2
    cos^2(x) = (adj)^2/(hyp)^2

    Adding them:

    sin^2(x) + cos^2(x) = \frac{(opp)^2 + (adj)^2}{(hyp)^2}

    By the definition of a right angled triangle and using Pythagoras the right side cancels to 1.

    ----------------

    2. Expand to give:

    cos^2(x) - 2sin(x)cos(x) + sin^2(x) + cos^2(x) + 2sin(x)cos(x) + cos^2(x)

    Remember what cos^2(x)+sin^2(x) is equal to

    ------------------

    3. Rewrite in terms of sin and cos:

    \frac{cos(x)}{sin(x)} + \frac{sin(x)}{cos(x)}

    Give them the same denominator by cross multiplying

    \frac{cos^2(x) + sin^2(x)}{sin(x)cos(x)}

    and cancel to give the rhs
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  3. #3
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    I know...here is the exact problem.

    What is wrong with the following proof?

    sin^2x + cos^2x = 1,
    sin^2x + cos^2x = sin^2x + (1 - sin^2x),
    sin^2x + (1 - sin^2x) = sin^2x - sin^2x + 1,
    = 1.

    Is there anything wrong with this?
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  4. #4
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    Quote Originally Posted by live_laugh_luv27 View Post
    I know...here is the exact problem.

    What is wrong with the following proof?

    sin^2x + cos^2x = 1,
    sin^2x + cos^2x = sin^2x + (1 - sin^2x),
    sin^2x + (1 - sin^2x) = sin^2x - sin^2x + 1,
    = 1.

    Is there anything wrong with this?
    Yeah, you're using the identity you're trying to prove as part of the proof (the bit in red). This shouldn't be done but I can't remember the name for it
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  5. #5
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    Quote Originally Posted by e^(i*pi) View Post
    Yeah, you're using the identity you're trying to prove as part of the proof (the bit in red). This shouldn't be done but I can't remember the name for it

    Oh, ok. So you can't use that part, because then you would be assuming you already proved the identity, which you didn't.
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    Post

    thanks for your help!

    Anyone have any idea how to start #2 and 3?
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  7. #7
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    Quote Originally Posted by live_laugh_luv27 View Post
    thanks for your help!

    Anyone have any idea how to start #2 and 3?
    Yeah, check out post 2
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  8. #8
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    oh wow...thanks
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