Editing Spherical trigonometry (section)

===Half-angle and half-side formulae===
With <math>2s=(a+b+c)</math> and <math>2S=(A+B+C),</math>
<math display="block">
\begin{alignat}{5}
  \sin{\tfrac{1}{2}}A &= \sqrt{\frac{\sin(s-b)\sin(s-c)}{\sin b\sin c}}
&\qquad\qquad
 \sin{\tfrac{1}{2}}a &= \sqrt{\frac{-\cos S\cos (S-A)}{\sin B\sin C}} \\[2ex]
 \cos{\tfrac{1}{2}}A &= \sqrt{\frac{\sin s\sin(s-a)}{\sin b\sin c}}
  & \cos{\tfrac{1}{2}}a &= \sqrt{\frac{\cos (S-B)\cos (S-C)}{\sin B\sin C}} \\[2ex]
  \tan{\tfrac{1}{2}}A &= \sqrt{\frac{\sin(s-b)\sin(s-c)}{\sin s\sin(s-a)}}
  & \tan{\tfrac{1}{2}}a &= \sqrt{\frac{-\cos S\cos (S-A)}{\cos (S-B)\cos(S-C)}}
 \end{alignat}
</math>

Another twelve identities follow by cyclic permutation.

The proof (Todhunter,<ref name=todhunter/> Art.49) of the first formula starts from the identity <math>2\sin^2\!\tfrac{A}{2} = 1 - \cos A,</math> using the cosine rule to express {{mvar|A}} in terms of the sides and replacing the sum of two cosines by a product. (See [[List of trigonometric identities#Product-to-sum and sum-to-product identities|sum-to-product identities]].) The second formula starts from the identity <math>2\cos^2\!\tfrac{A}{2} = 1 + \cos A,</math> the third is a quotient and the remainder follow by applying the results to the polar triangle.