![{\displaystyle \operatorname {Sin} .s={\frac {\operatorname {Sin} .b\operatorname {Cos} .{\frac {1}{2}}C\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}B}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B}{\operatorname {Sin} .{\frac {1}{2}}C}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e443d2b9d894d60bea2f6587877488c83bdabc82)
(
xxiv)
![{\displaystyle -\operatorname {Cos} .S={\frac {\operatorname {Sin} .B\operatorname {Sin} .{\frac {1}{2}}c\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Cos} .{\frac {1}{2}}b}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b}{\operatorname {Cos} .{\frac {1}{2}}c}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b204f3484c389a3599c8d2147fae04b54d99c7)
(
XXIV)
![{\displaystyle \operatorname {Sin} .(s-a)={\frac {\operatorname {Sin} .b\operatorname {Sin} .{\frac {1}{2}}C\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Cos} .{\frac {1}{2}}B}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B}{\operatorname {Cos} .{\frac {1}{2}}C}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e75918fe050ae6eb1b8f80739211553e1dfd23b)
(
xxv)
![{\displaystyle \operatorname {Cos} .(S-A)={\frac {\operatorname {Sin} .B\operatorname {Cos} .{\frac {1}{2}}c\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Sin} .{\frac {1}{2}}b}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b}{\operatorname {Sin} .{\frac {1}{2}}c}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df7301bc717cb3ac6311f4c0856f34ca2589232)
(
XXV)
En vertu des formules (29) et (30), les valeurs (vii) et (VII) peuvent être écrites ainsi
![{\displaystyle p^{2}={\frac {1}{4}}\left(1-\operatorname {Cos} .^{2}a-\operatorname {Cos} .^{2}b-\operatorname {Cos} .^{2}c+2\operatorname {Cos} .a\operatorname {Cos} .b\operatorname {Cos} .c\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/940e1c8679864dbdb353c8103a0b54fabd67e92d)
(
xxvi)
![{\displaystyle P^{2}={\frac {1}{4}}\left(1-\operatorname {Cos} .^{2}A-\operatorname {Cos} .^{2}B-\operatorname {Cos} .^{2}C-2\operatorname {Cos} .A\operatorname {Cos} .B\operatorname {Cos} .C\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f8264b153ed66785889dc0059ba787c973f0fe)
(
XXVI)
En transformant les cosinus d’angles en cosinus de moitiés, dans la première et en sinus de moitiés dans la seconde, à l’aide des formules (5), on trouvera
![{\displaystyle p^{2}=4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Cos} .^{2}{\frac {1}{2}}b\operatorname {Cos} .^{2}{\frac {1}{2}}c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7994d2298726e31f0f27cce6547c621e86a004df)
![{\displaystyle -\left(1-\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c\right)^{2},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c836b4f0e78c470bb54750f8576098d9cff2ed08)
(
xxvii)
![{\displaystyle P^{2}=4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Sin} .^{2}{\frac {1}{2}}B\operatorname {Sin} .^{2}{\frac {1}{2}}C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317d370ee9cd592b9e914d65fa4d3dc039c28e3d)
![{\displaystyle -\left(1-\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C\right)^{2}\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbac81a186c4e6d2b5d60c147314291363e1e160)
(
XXVII)
ou encore
![{\displaystyle p^{2}=4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Sin} .^{2}{\frac {1}{2}}b\operatorname {Sin} .^{2}{\frac {1}{2}}c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/698715bc50c22f42a1adc2867f4a5742aa3ad43d)
![{\displaystyle -\left(1+\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c\right)^{2},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0bfc5294dc845b3cad767da3d9cfcf951f2570)
(
xxviii)
![{\displaystyle P^{2}=4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Cos} .^{2}{\frac {1}{2}}B\operatorname {Cos} .^{2}{\frac {1}{2}}C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9a598032d24d66df64bebe9b6f24fc7de7ad4c)
![{\displaystyle -\left(1+\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C\right)^{2}.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf16d9ef713d88a323cbdc34e34619f98569106)
(
XXVIII)
Cela posé, on a, par les formules (xvii) et (XVII),