![{\displaystyle \operatorname {Cos} .^{2}(r+\mathrm {D} )-\operatorname {Cos} .^{2}\mathrm {R} =\operatorname {Sin} .(\mathrm {R} +r+\mathrm {D} )\operatorname {Sin} .(\mathrm {R} -r-\mathrm {D} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89002129a80b3f83d186ba47dd11a599e4e2aefc)
![{\displaystyle \operatorname {Sin} .^{2}(R+\mathrm {D} )-\operatorname {Sin} .^{2}\mathrm {r} =\operatorname {Sin} .(\mathrm {R} +r+\mathrm {D} )\operatorname {Sin} .(\mathrm {R} -r-\mathrm {D} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415917d551f839b45c21f85fac360c92ab73ac27)
mettant donc ces valeors et supprimant le facteur
facteur commun d’abord aux deux termes du premier rapport ; puis aux deux autecédans, il viendra
![{\displaystyle \operatorname {Cos} .^{2}\mathrm {R} :\operatorname {Sin} .(\mathrm {R} -r-\mathrm {D} )::\operatorname {Sin} .(\mathrm {R} -r+\mathrm {D} ):\operatorname {Sin} .^{2}r\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82a680a0f42335886dfd989c8c0d34b98467e813)
d’où, en égalant le produit des extrêmes à celui des moyens,
![{\displaystyle \operatorname {Sin} .^{2}r\operatorname {Cos} .^{2}\mathrm {R} =\operatorname {Sin} ^{2}.(\mathrm {R} -r)-\operatorname {Sin} .^{2}\mathrm {D} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37540537687bf13e2f9c3e278c7b9cb595216225)
c’est-à-dire,
![{\displaystyle \operatorname {Sin} .^{2}\mathrm {D} =\operatorname {Sin} ^{2}.(\mathrm {R} -r)-\operatorname {Sin} .^{2}r\operatorname {Cos} .^{2}\mathrm {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed93c20053eaead6d8222be09087107e3a11a4a)
ou
![{\displaystyle \operatorname {Sin} .^{2}\mathrm {D} =\left\{\operatorname {Sin} .(\mathrm {R} -r)+\operatorname {Sin} .r\operatorname {Cos} .\mathrm {R} \right\}\left\{\operatorname {Sin} .(\mathrm {R} -r)-\operatorname {Sin} .r\operatorname {Cos} .\mathrm {R} \right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f085f3aad8eef5653ed717e7facfe04f7055ac46)
mais
![{\displaystyle \operatorname {Sin} .r\operatorname {Cos} .\mathrm {R} ={\frac {\operatorname {Sin} .(\mathrm {R} +r)-\operatorname {Sin} .(\mathrm {R} -r)}{2}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3597576127510909c757c968bc6b74e527090622)
donc, finalement,
![{\displaystyle \operatorname {Sin} .^{2}\mathrm {D} ={\frac {\operatorname {Sin} .(\mathrm {R} +r)+\operatorname {Sin} .(\mathrm {R} -r)}{2}}.{\frac {3\operatorname {Sin} .(\mathrm {R} -r)-\operatorname {Sin} .(\mathrm {R} +r)}{2}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a82b86f1f0b06c7870c5d126d0c9e2a6cfc0dc45)
c’est-à-dire,