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The not-so-well-known triple tangent and triple cotangent identities,

If $x + y + z = \pi$ then $\tan x + \tan y + \tan z = \tan x\tan y\tan z \;\;\; (x,y,z \neq \pi/2+\pi n)$.

If $x + y + z = \frac\pi 2$ then $\cot x + \cot y + \cot z = \cot x \cot y \cot z \;\;\; (x,y,z \neq \pi n)$.

are usually proved analytically. Are there geometric proofs of these identities? Or at least geometric interpretations that might provide some intuition for why they are true?

krazy-8
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Ted Hopp
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  • I don't know if there is a geometric way to show this, but I'll leave an algebraic solution: https://www.askiitians.com/forums/Trigonometry/if-x-y-z-pi-prove-cotx-2-coty-2-cotx-2-cotx-2_254179.htm – Prometheus May 21 '21 at 02:16
  • @Prometheus - Thanks. There are plenty of analytic proofs floating around the web, including a Wikipedia article with proofs of these and many other trig identities. It has a nice geometric proof of the angle sum identities for sine and cosine, but unfortunately nothing like that for the identities I asked about. – Ted Hopp May 21 '21 at 03:09

1 Answers1

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trigonograph

Let $AF=\tan\theta$. Then, obtain expressions for $AE, EF, EC, EB, BC, FD,$ and $DC$ in order.

$\implies AD=BC \;\; \blacksquare$

trigonograph for cot

Similarly, let $FD=\cot\theta$, and let the magic happen...

krazy-8
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  • Nice! This is exactly the kind of thing I was looking for. I'm going to try to use it as inspiration to try developing a geometric proof of the cotangent identity as well. Unless you have that one hanging around as well? – Ted Hopp May 21 '21 at 13:31
  • @TedHopp I have also added a diagram for cotangent, hope you like it. – krazy-8 May 22 '21 at 05:18
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    Thanks again. I hadn't had a chance to try my hand at a proof, and now I don't need to. :) – Ted Hopp May 23 '21 at 01:20