TITLE: What are some interesting and unusual theorems, identities, notations in Trigonometry?
QUESTION [6 upvotes]: Context/Motivation
When studying a subject I usually like to create a document and make a kind of study guide, notes about the subject. I created the document and write using LaTeX. Also, besides learning about the subject itself (it is math or something related to robotics) I practice LaTeX and English because I write it in English.  Taking trigonometry, the topic of this question, I would write lecturing myself and go through concepts, theorems, formulas detailing as much as possible. Yesterday night I was wondering about making notes about trigonometry and it reminded me of a notation that I did find unusual when I first saw it, but it was interesting regarding the Domain of the tangent function.
Working with trigonometry function such that $f:\mathbb{R}\rightarrow\mathbb{R}$, we have $f(x)=\tan(x)=\dfrac{\sin(x)}{\cos(x)}$. As $\cos(x)\neq 0$ the domain of the tangent function is usually written as $\operatorname{Dom}(\tan) = \left\{x: x \neq \dfrac{\pi}{2}+k\pi, k \in \mathbb{Z} \right\}$.
We have $\cos(x) = 0$  for $x=\pm\dfrac{\pi}{2}, \pm\dfrac{3\pi}{2}, \pm\dfrac{5\pi}{2}...$
In fact, taking the intervals we have
$$... \quad \cup\left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right)\cup \quad ...$$
For $k\in\mathbb{Z}$. Considering $x_k = \dfrac{\pi}{2}+k\pi = \dfrac{2k\pi+\pi}{2}= \boxed{\dfrac{(2k+1)\pi}{2}}$, we can note that the intervals are determined by these $x_k$ points, so
$$(x_k, x_{k+1})= \left(\dfrac{(2k+1)\pi}{2}, \dfrac{(2k+3)\pi}{2}\right)$$
Once we have an infinity quantity of unions, we can represent it using the big union notation. This is a notation I was not familiar, that is why I found it very interesting. Therefore, we have
$$\operatorname{Dom}(\tan) = \left\{x: x \neq \dfrac{\pi}{2}+k\pi, k \in \mathbb{Z} \right\} = \bigcup_{k=-\infty}^{\infty} \left(\dfrac{(2k+1)\pi}{2}, \dfrac{(2k+3)\pi}{2}\right)$$
Question
I'm afraid this question might be off-topic but I am asking this here because it seems to be a topic where experience is very important. I am looking identities, theorems, concepts, notations that are not very common, usual in regular books, textbooks. It is just out of my curiosity, but I am really interested in these new things. I'm sure the example I gave above might not be impressive for university students or most people who use MSE, but I was very surprised when I first saw it.
More precisely, I am looking for answers of the following kind:

Trigonometric Identities that are unusual that might/might not have interesting applications;
Notations that are not very common;
Concepts, Theorems that are usually not seem is regular high-school but actually presents interesting ideas;
University trigonometry concepts with interesting ideas/application in high schools trigonometry;

The question is certainly very wide regarding the topic, but any interesting information is very welcomed.

REPLY [2 votes]: Trigonometry is based on relationships between
the side lengths $a,b,c$
and corresponding angles $\alpha,\beta,\gamma$
of triangle,
but sometimes it is more convenient to use
express these relations in terms of
another three linear properties of triangle,
namely semiperimeter $\rho=\tfrac12(a+b+c)$,
inradius $r$ and circumradius $R$ of the triangle.
It is especially useful, that trigonometric expressions for the angles
can be formulated in terms of just two unitless parameters,
$u=\tfrac\rho{R}$ and $v=\tfrac r{R}$,
which can be considered as the semiperimeter
and the inradius of triangle, inscribed in the unit circle ($R=1$).
This is useful since the ranges of these two parameters $u,v$,
that correspond to a valid triangle (including degenerate cases),
are bounded as
\begin{align} 
v&\in[0,\tfrac12]
,\quad u(v)\in[u_{\min}(v),u_{\max}(v)]
\tag{1}\label{1}
,\\
u_{\min}(v)&=\sqrt{27-(5-v)^2-2\sqrt{(1-2v)^3}}
\tag{2}\label{2}
,\\
u_{\max}(v)&=\sqrt{27-(5-v)^2+2\sqrt{(1-2v)^3}}
\tag{3}\label{3}
.
\end{align}
Curves \eqref{2}, \eqref{3} and the line $v=0$ bound the region,
for which any pair $(v,u)$ correspond to unique triangle,
that is a triple of angles $\alpha,\beta,\gamma$.
Any pair $(v,u)$ outside that region can not define
(be a properties of) a valid triangle.
The boundary curves $u_{\min}(v)$, $u_{\max}(v)$
correspond to isosceles triangles,
the line $u(v)=v+2$ inside the region represent right-angled triangles,
the point where
$u_{\min}(v)=u_{\max}(v)=u_{\min}(\tfrac12)=u_{\max}(\tfrac12)=\tfrac32\sqrt3$
corresponds to equilateral triangle.
Not surprisingly, any triplet of the trigonometric functions
of $\alpha,\beta,\gamma$ can be found as a solution to the cubic equation
\begin{align} 
x^3-a_2x^2+a_1x-a_0&=0
\tag{4}\label{4}
,
\end{align}
where $a_i$ can be expressed in terms of $u,v$.
For $\cos(\cdot)$ we have:
\begin{align}
a_0&=\cos\alpha\cos\beta\cos\gamma
=\tfrac14(u^2-(v+2)^2)
\tag{5}\label{5}
,\\
a_1&=
\cos\alpha\cos\beta+\cos\beta\cos\gamma+\cos\alpha\cos\gamma
=\tfrac14(u^2+v^2)-1
\tag{6}\label{6}
\\
a_2&=\cos\alpha+\cos\beta+\cos\gamma
=v+1
\tag{7}\label{7}
.
\end{align}
For $\cot(\cdot)$ we have:
\begin{align}
a_0&=\cot\alpha\cot\beta\cot\gamma
=\frac{u^2-(v+2)^2}{2u\,v}
\tag{8}\label{8}
,\\
a_1&=\cot\alpha\cot\beta+\cot\beta\cot\gamma+\cot\gamma\cot\alpha
=1
\tag{9}\label{9}
,\\
a_2&=\cot\alpha+\cot\beta+\cot\gamma
=
\tfrac12\,\left(\frac{u}{v}-\frac{v}{u}\right)-\frac{2}{u}
\tag{10}\label{10}
,
\end{align}
and so on.