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Solve this equation algebraically:

(1-sin x)/cos x = cos x/(1+sin x)

I know the answer is an identity, and when graphed, it looks like cot x. I just don't know how to get there. I tried multiplying each side by its conjugate, but I still feel stuck. This is what I have so far:

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

...but I'm not really sure how to get to the answer. Help please?
Thank you!

  • Math -

    Do you mean solve it or prove it?

    It is an identity, so there really isn't a specific solution: it's true for all x.

    I suggest you try reformatting as

    (1-sin x)/cos x - cos x/(1+sin x) = 0

    Then bring both fractions to the common denominator (cosx)(1+sinx), and I think you'll recognise the numerator you're left with!

  • Math -

    Ok, I worked it out, and so far i have this:

    cos^2x - cos x - sinxcosx /
    cosx + sinxcosx

    Now I just need some help with reducing?

  • Math -

    Um, no. Wrong turn somewhere. It's a LOT simpler than that.

    Your numerator will be :

    (1-sinx)(1+sinx) - cos^2x

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