Теорема тангенсов

\[{\frac {a-b}{a+b}}={\frac {{\mathop {\operatorname {tg} }}\left[{\tfrac {1}{2}}(A-B)\right]}{{\mathop {\operatorname {tg} }}\left[{\tfrac {1}{2}}(A+B)\right]}}\]

\[{\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}\]

\[{d = \frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}\]

\[{\displaystyle a=d\sin \alpha }\]

\[{\displaystyle b=d\sin \beta }\]

\[{\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}\]

\[{\displaystyle \sin \alpha \pm \sin \beta =2\sin {\frac {\alpha \pm \beta }{2}}\cos {\frac {\alpha \mp \beta }{2}}}\]

\[{\displaystyle {\frac {a-b}{a+b}}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}={\frac {2\sin {\frac {\alpha -\beta }{2}}\cos {\frac {\alpha +\beta }{2}}}{2\sin {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}}}={\frac {\mathrm {tg} {\frac {\alpha -\beta }{2}}}{\mathrm {tg} {\frac {\alpha +\beta }{2}}}}}\]