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 :(

Attachment image

5 Answers

Relevance
  • nbsale
    Lv 6
    8 months ago
    Favorite 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.

    • Commenter avatarLogin to reply the answers
  • Pope
    Lv 7
    8 months 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.

    • ...Show all comments
    • Pope
      Lv 7
      8 months 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.

    • Commenter avatarLogin to reply the answers
  • Ray S
    Lv 7
    8 months ago

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

    = 1/(1 − tanθ)

    • Commenter avatarLogin to reply the answers
  • ted s
    Lv 7
    8 months ago

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

    • Commenter avatarLogin to reply the answers
  • How do you think about the answers? You can sign in to vote the answer.
  • 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))

    • Commenter avatarLogin to reply the answers
Still have questions? Get your answers by asking now.