Trig help!!?

I need to prove this equation to be true (establishing the identity). I’ve tried everything that I could think of and feel like I’m missing something obvious :(

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  • nbsale
    Lv 6
    4 weeks ago
    Best Answer

    Equations like this always have limitations since the denominators can be zero, and tan isn't defined for all values. But what you are looking at is easy to prove for where it's defined.

    Assuming cosΘ is not zero (i.e., Θ <> π/2 + nπ), divide the numerator and denominator of the left hand side by cosΘ.

    You get

    (cos/cos) / (cos/cos - sin/cos)

    = 1/(1 - tan) QED

    It's also not defined for values where Θ = π/4 + nπ, i.e., 45 degrees, or 225 degrees, or their equivalents.

  • Pope
    Lv 7
    4 weeks ago

    The equation is not an identity.

    Let θ = π/2.

    LHS

    = cos(π/2) / [cos(π/2) - sin(π/2)]

    = 0 / (0 - 1)

    = 0

    Since tan(π/2) is undefined, so is the right side of the equation. The purported identity equates a real number with an undefined expression. It is not an identity at all.

    • Pope
      Lv 7
      4 weeks agoReport

      @ nbsale

      Yes, I agree that the two sides are equal wherever they are both defined, and RHS does approach LHS at all of the trouble spots. However, neither of those properties make the equation an identity.

      Those exclusions you suggested in your own answer would suffice, but they were not given.

  • Ray S
    Lv 7
    4 weeks ago

       cosθ/(cosθ − sinθ)       ← divide top and bottom by cosθ

    = 1/(1 − tanθ)

  • ted s
    Lv 7
    4 weeks ago

    work on the right side since tan = sin / cos....2 steps

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  • Try factoring out cos(t)

    cos(t) / (cos(t) - sin(t)) =>

    cos(t) * 1 / (cos(t) * (1 - sin(t)/cos(t))) =>

    1 / (1 - sin(t)/cos(t)) =>

    1 / (1 - tan(t))

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