Le théorème déjà cité de M. Cotes nous donne l’équation
![{\displaystyle {\begin{aligned}p^{2m}-q^{2m}&=\left(p^{2}-q^{2}\right)\left(p^{2}-2pq\cos {\frac {\varpi }{2m}}+q^{2}\right)\left(p^{2}-2pq\cos {\frac {2\varpi }{2m}}+q^{2}\right)\\&\times \left(p^{2}-2pq\cos {\frac {3\varpi }{2m}}+q^{2}\right)\ldots \left[p^{2}-2pq\cos {\frac {(m-1)\varpi }{2m}}+q^{2}\right],\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ebe5421daa9e257c3fe13289827701f4c3c529)
en n’omettant aucun des facteurs intermédiaires ; que l’on compare donc ces facteurs avec ceux de l’équation précédente, en faisant
![{\displaystyle p^{2}+q^{2}=\cos {\frac {\lambda \varpi }{2m}},\qquad 2pq=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/933f6ad26796db1aa8b9ae79b5700e877f5de8a7)
et l’on aura
![{\displaystyle {\begin{aligned}p^{2}+2pq+q^{2}=&1+\cos {\frac {\lambda \varpi }{2m}}=2\left(\cos {\frac {\lambda \varpi }{4m}}\right),\\p^{2}-2pq+q^{2}=&\cos {\frac {\lambda \varpi }{2m}}-1=-2\left(\sin {\frac {\lambda \varpi }{4m}}\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3b17ea3e167846868712af0d487252d1819099)
d’où, en extrayant les racines, il résulte
![{\displaystyle {\begin{aligned}p+q=&\pm {\sqrt {2}}\cos {\frac {\lambda \varpi }{2m}},\\p-q=&\pm {\sqrt {2}}\sin {\frac {\lambda \varpi }{2m}}{\sqrt {-1}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595540ae67b20bc39ed9e861a7856b5e7810b2d0)
et enfin
![{\displaystyle {\begin{aligned}p=&\pm {\frac {\cos {\cfrac {\lambda \varpi }{4m}}+\sin {\cfrac {\lambda \varpi }{4m}}{\sqrt {-1}}}{\sqrt {2}}},\\q=&\pm {\frac {\cos {\cfrac {\lambda \varpi }{4m}}-\sin {\cfrac {\lambda \varpi }{4m}}{\sqrt {-1}}}{\sqrt {2}}},\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d158946c15ef4ced26ff3c8f8106bf54a9d9e2fa)
Par conséquent on aura
![{\displaystyle {\begin{aligned}p^{2}=&{\frac {1}{2}}\left(\cos {\frac {\lambda \varpi }{4m}}+\sin {\frac {\lambda \varpi }{4m}}{\sqrt {-1}}\right)^{2}={\frac {1}{2}}\left(\cos {\frac {\lambda \varpi }{2m}}+\sin {\frac {\lambda \varpi }{2m}}{\sqrt {-1}}\right),\\q^{2}=&{\frac {1}{2}}\left(\cos {\frac {\lambda \varpi }{4m}}-\sin {\frac {\lambda \varpi }{4m}}{\sqrt {-1}}\right)^{2}={\frac {1}{2}}\left(\cos {\frac {\lambda \varpi }{2m}}-\sin {\frac {\lambda \varpi }{2m}}{\sqrt {-1}}\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa94d2725e23992fddbf395d4884193516febb71)
![{\displaystyle p^{2}-q^{2}=\sin {\frac {\lambda \varpi }{2m}}{\sqrt {-1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd9b1e378a54968c4fff3a6b3ffc9adefea3f2a)