Mollweide's formula

Because in a planar triangle ${\displaystyle {\tfrac {1}{2}}\gamma ={\tfrac {1}{2}}\pi -{\tfrac {1}{2}}(\alpha +\beta ),}$ these identities can alternately be written in a form in which they are more clearly a limiting case of Napier's analogies for spherical triangles,

${\displaystyle {\begin{aligned}{\frac {a+b}{c}}&={\frac {\cos {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}(\alpha +\beta )}},\\[10mu]{\frac {a-b}{c}}&={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\sin {\tfrac {1}{2}}(\alpha +\beta )}}.\end{aligned}}}$

Dividing one by the other to eliminate ${\textstyle c}$ results in the law of tangents,

${\displaystyle {\begin{aligned}{\frac {a+b}{a-b}}={\frac {\tan {\tfrac {1}{2}}(\alpha +\beta )}{\tan {\tfrac {1}{2}}(\alpha -\beta )}}.\end{aligned}}}$

In terms of half-angle tangents alone, Mollweide's formula can be written as

${\displaystyle {\begin{aligned}{\frac {a+b}{c}}&={\frac {1+\tan {\tfrac {1}{2}}\alpha \,\tan {\tfrac {1}{2}}\beta }{1-\tan {\tfrac {1}{2}}\alpha \,\tan {\tfrac {1}{2}}\beta }},\\[10mu]{\frac {a-b}{c}}&={\frac {\tan {\tfrac {1}{2}}\alpha -\tan {\tfrac {1}{2}}\beta }{\tan {\tfrac {1}{2}}\alpha +\tan {\tfrac {1}{2}}\beta }},\end{aligned}}}$

or equivalently

${\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}\alpha \,\tan {\tfrac {1}{2}}\beta &={\frac {a+b-c}{a+b+c}},\\[10mu]{\frac {\tan {\tfrac {1}{2}}\alpha }{\tan {\tfrac {1}{2}}\beta }}&={\frac {{\phantom {-}}a-b+c}{-a+b+c}}.\end{aligned}}}$

Multiplying the respective sides of these identities gives one half-angle tangent in terms of the three sides,

${\displaystyle {\bigl (}\tan {\tfrac {1}{2}}\alpha {\bigr )}^{2}={\frac {(a+b-c)(a-b+c)}{(a+b+c)(-a+b+c)}}.}$

which becomes the law of cotangents after taking the square root,

${\displaystyle {\frac {\cot {\tfrac {1}{2}}\alpha }{s-a}}={\frac {\cot {\tfrac {1}{2}}\beta }{s-b}}={\frac {\cot {\tfrac {1}{2}}\gamma }{s-c}}={\sqrt {\frac {s{\vphantom {)}}}{(s-c)(s-b)(s-a)}}},}$

where ${\textstyle s={\tfrac {1}{2}}(a+b+c)}$ is the semiperimeter.

The identities can also be proven equivalent to the law of sines and law of cosines.

In spherical trigonometry, the law of cosines and derived identities such as Napier's analogies have precise duals swapping central angles measuring the sides and dihedral angles at the vertices. In the infinitesimal limit, the law of cosines for sides reduces to the planar law of cosines and two of Napier's analogies reduce to Mollweide's formulas above. But the law of cosines for angles degenerates to ${\displaystyle 0=0.}$ By dividing squared side length by the spherical excess ${\displaystyle E,}$ we obtain a non-vanishing ratio, the spherical trigonometry relation:

${\displaystyle {\begin{aligned}{\frac {\tan ^{2}{\tfrac {1}{2}}c}{\tan {\tfrac {1}{2}}E}}={\frac {\sin \gamma }{\sin \alpha \,\sin \beta }}.\end{aligned}}}$

In the infinitesimal limit, as the half-angle tangents of spherical sides reduce to lengths of planar sides, the half-angle tangent of spherical excess reduces to twice the area ${\displaystyle A}$ of a planar triangle, so on the plane this is:

${\displaystyle {\frac {c^{2}}{2A}}={\frac {\sin \gamma }{\sin \alpha \,\sin \beta }},}$

and likewise for ${\displaystyle a}$ and ${\displaystyle b.}$

As corollaries (multiplying or dividing the above formula in terms of ${\displaystyle a}$ and ${\displaystyle b}$) we obtain two dual statements to Mollweide's formulas. The first expresses the area in terms of two sides and the included angle, and the other is the law of sines:

${\displaystyle {\frac {ab}{2A}}={\frac {1}{\sin \gamma }},}$

${\displaystyle {\frac {a}{b}}={\frac {\sin \alpha }{\sin \beta }}.}$

We can alternately express the second formula in a form closer to one of Mollweide's formulas (again the law of tangents):

${\displaystyle {\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\cot {\tfrac {1}{2}}\gamma }}={\frac {a-b}{a+b}}.}$

• H. Arthur De Kleine, "Proof Without Words: Mollweide's Equation", Mathematics Magazine, volume 61, number 5, page 281, December, 1988.