# 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
8 months ago

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
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.

• 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.

• Ray S
Lv 7
8 months ago

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

= 1/(1 − tanθ)

• ted s
Lv 7
8 months ago

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

• 8 months ago

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))