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Math Help - Can't prove these identities?

  1. #1
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    Can't prove these identities?

    1) sin(x+y)sin(x-y) = cos^2 y - cos^2 x
    (sin x cos y + cos x sin y)(sin x cos y - cos x sin y) = cos^2 y - cos^2 x

    From there I don't know where to go...

    2) tan(pi/4 + x) + tan(pi/4 - x) = 2sec2x

    For this one I don't even know where to start, I can't even find a formula that I could use for the LS.

    All and any help is very much welcomed!
    Thanks in advance
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  2. #2
    MHF Contributor Unknown008's Avatar
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    1) sin(x+y)sin(x-y) = cos^2 y - cos^2 x
    (sin x cos y + cos x sin y)(sin x cos y - cos x sin y) = cos^2 y - cos^2 x

    From there I don't know where to go...
    From there, use the difference of 2 squares.

    Then, use the Pythagorean identity to substitute the sin by cos and you'll get it.

    2. Start by using the compound angle identity for tan!
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  3. #3
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    Quote Originally Posted by doodofnood View Post
    1) sin(x+y)sin(x-y) = cos^2 y - cos^2 x
    (sin x cos y + cos x sin y)(sin x cos y - cos x sin y) = cos^2 y - cos^2 x
    (a+b)(a-b) = a^2-b^2 (difference of two squares).
    2) tan(pi/4 + x) + tan(pi/4 - x) = 2sec2x
    For this one I don't even know where to start, I can't even find a formula that I could use for the LS.
    \displaystyle \tan(x+y) = \frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}

    Oh, late. Ah, well...
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  4. #4
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    so for the first 1 i got down to
    sin^2xcos^2y - cos^2xsin^2y
    = (1-cos^2x)cos^y - cos^2x(1-cos^2y)

    now what...?
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  5. #5
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    Quote Originally Posted by doodofnood View Post
    now what...?
    Well, why did you stop? Keep going, expand the little barackets.
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  6. #6
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    so if i expand i get..

    cos^2y - cos^4xy - cos^2x - cos^4xy
    then cos^4xy cancels out?
    :O
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  7. #7
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by doodofnood View Post
    so if i expand i get..

    cos^2y - cos^4xy - cos^2x - cos^4xy
    then cos^4xy cancels out?
    :O
    you made a small mistake:

    (1-cos^2x)cos^2y - cos^2x(1-cos^2y)

    = cos^2y -cos^4y-cos^2x+cos^4y

    now the cos^4y cancel out and you are left with..
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