Page:Joseph Louis de Lagrange - Œuvres, Tome 7.djvu/355

donc, ajoutant ensemble, on aura

$\mathrm {\left(\operatorname {tang} {\frac {A}{2}}+\operatorname {tang} {\frac {B}{2}}\right)\operatorname {tang} {\frac {C}{2}}} ={\frac {\sin {\cfrac {a-b+c}{2}}+\sin {\cfrac {b+c-a}{2}}}{\sin {\cfrac {a+b+c}{2}}}}$

$={\frac {2\sin {\cfrac {c}{2}}\cos {\cfrac {a-b}{2}}}{\sin {\cfrac {a+b+c}{2}}}},$

et, retranchant les mêmes équations l’une de l’autre, on aura

$\mathrm {\left(\operatorname {tang} {\frac {A}{2}}-\operatorname {tang} {\frac {B}{2}}\right)\operatorname {tang} {\frac {C}{2}}} ={\frac {\sin {\cfrac {a-b+c}{2}}-\sin {\cfrac {b+c-a}{2}}}{\sin {\cfrac {a+b+c}{2}}}}$

$={\frac {2\cos {\cfrac {c}{2}}\sin {\cfrac {a-b}{2}}}{\sin {\cfrac {a+b+c}{2}}}}.$

D’un autre côté, on a

$1+\mathrm {\operatorname {tang} {\frac {A}{2}}\operatorname {tang} {\frac {B}{2}}} ={\frac {\sin {\cfrac {a+b+c}{2}}+\sin {\cfrac {a+b-c}{2}}}{\sin {\cfrac {a+b+c}{2}}}}={\frac {2\sin {\cfrac {a+b}{2}}\cos {\cfrac {c}{2}}}{\sin {\cfrac {a+b+c}{2}}}},$

$1-\mathrm {\operatorname {tang} {\frac {A}{2}}\operatorname {tang} {\frac {B}{2}}} ={\frac {\sin {\cfrac {a+b+c}{2}}-\sin {\cfrac {a+b-c}{2}}}{\sin {\cfrac {a+b+c}{2}}}}={\frac {2\sin {\cfrac {c}{2}}\cos {\cfrac {a+b}{2}}}{\sin {\cfrac {a+b+c}{2}}}}.$

Donc, puisque $\mathrm {\operatorname {tang} {\frac {A\pm B}{2}}={\frac {\operatorname {tang} {\cfrac {A}{2}}\pm \operatorname {tang} {\cfrac {B}{2}}}{1\mp \operatorname {tang} {\cfrac {A}{2}}\operatorname {tang} {\cfrac {B}{2}}}}} ,$ on aura ces deux équations

$(r)\qquad \qquad \qquad \qquad \mathrm {\operatorname {tang} {\frac {A+B}{2}}\operatorname {tang} {\frac {C}{2}}} ={\frac {\cos {\cfrac {a-b}{2}}}{\cos {\cfrac {a+b}{2}}}},$

$(s)\qquad \qquad \qquad \qquad \mathrm {\operatorname {tang} {\frac {A-B}{2}}\operatorname {tang} {\frac {C}{2}}} ={\frac {\sin {\cfrac {a-b}{2}}}{\sin {\cfrac {a+b}{2}}}}.$

On peut déduire des formules semblables de l’équation $(n)$ du n^o 24, et, sans faire un nouveau calcul, il n’y aura qu’à changer les côtés $a,b,c$ dans les suppléments à deux droits des angles $\mathrm {A,B,C} ,$ et ces angles