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 :(
- nbsaleLv 68 months agoFavorite 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.
(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.
- PopeLv 78 months ago
The equation is not an identity.
Let θ = π/2.
= cos(π/2) / [cos(π/2) - sin(π/2)]
= 0 / (0 - 1)
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.
- Ray SLv 78 months ago
cosθ/(cosθ − sinθ) ← divide top and bottom by cosθ
= 1/(1 − tanθ)
- ted sLv 78 months ago
work on the right side since tan = sin / cos....2 steps
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- 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))