Editing Spherical trigonometry (section)

===Cotangent four-part formulae===
The six parts of a triangle may be written in cyclic order as ({{mvar|aCbAcB}}). The cotangent, or four-part, formulae relate two sides and two angles forming four ''consecutive'' parts around the triangle, for example ({{mvar|aCbA}}) or {{mvar|BaCb}}). In such a set there are inner and outer parts: for example in the set ({{mvar|BaCb}}) the inner angle is {{mvar|C}}, the inner side is {{mvar|a}}, the outer angle is {{mvar|B}}, the outer side is {{mvar|b}}. The cotangent rule may be written as (Todhunter,<ref name=todhunter/> Art.44)
<math display=block>
  \cos\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{side}\end{smallmatrix}\Bigr)
  \cos\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{angle}\end{smallmatrix}\Bigr) =
  \cot\!\Bigl(\begin{smallmatrix}\text{outer} \\ \text{side}\end{smallmatrix}\Bigr)
  \sin\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{side}\end{smallmatrix}\Bigr) -
  \cot\!\Bigl(\begin{smallmatrix}\text{outer} \\ \text{angle}\end{smallmatrix}\Bigr)
  \sin\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{angle}\end{smallmatrix}\Bigr),
</math>
and the six possible equations are (with the relevant set shown at right):
<math display=block>\begin{alignat}{5}
  \text{(CT1)}&& \qquad \cos b\,\cos C &= \cot a\,\sin b - \cot A \,\sin C \qquad&&(aCbA)\\[0ex]
  \text{(CT2)}&& \cos b\,\cos A &= \cot c\,\sin b - \cot C \,\sin A &&(CbAc)\\[0ex]
  \text{(CT3)}&& \cos c\,\cos A &= \cot b\,\sin c - \cot B \,\sin A &&(bAcB)\\[0ex]
  \text{(CT4)}&& \cos c\,\cos B &= \cot a\,\sin c - \cot A \,\sin B &&(AcBa)\\[0ex]
  \text{(CT5)}&& \cos a\,\cos B &= \cot c\,\sin a - \cot C \,\sin B &&(cBaC)\\[0ex]
  \text{(CT6)}&& \cos a\,\cos C &= \cot b\,\sin a - \cot B \,\sin C &&(BaCb)
\end{alignat}</math>
To prove the first formula start from the first cosine rule and on the right-hand side substitute for {{math|cos ''c''}} from the third cosine rule:
<math display=block>\begin{align}
  \cos a &= \cos b \cos c + \sin b \sin c \cos A \\
  &= \cos b\ (\cos a \cos b + \sin a \sin b \cos C) + \sin b \sin C \sin a \cot A \\
  \cos a \sin^2 b &= \cos b \sin a \sin b \cos C + \sin b \sin C \sin a \cot A.
\end{align}</math>
The result follows on dividing by {{math|sin ''a'' sin ''b''}}. Similar techniques
with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.