Origem: Wikilivros, livros abertos por um mundo aberto.
As fórmulas de transformação de soma e diferença em produto, também conhecidas como Fórmulas de Prostaférese[1], são:
![{\displaystyle \mathrm {sen} \,(x)+\mathrm {sen} \,(y)=2\cdot \mathrm {sen} \,\left({\frac {x+y}{2}}\right)\mathrm {cos} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/771ff37894b920b252c93df056d0b2940b8758f3)
![{\displaystyle \mathrm {sen} \,(x)-\mathrm {sen} \,(y)=2\cdot \mathrm {sen} \,\left({\frac {x-y}{2}}\right)\mathrm {cos} \,\left({\frac {x+y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83150a9cb4cefdc3d6281343f2c750c5f564661)
![{\displaystyle \mathrm {cos} \,(x)+\mathrm {cos} \,(y)=2\cdot \mathrm {cos} \,\left({\frac {x+y}{2}}\right)\mathrm {cos} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b976721df079c112155b51c4209de3c8ca1efd45)
![{\displaystyle \mathrm {cos} \,(x)-\mathrm {cos} \,(y)=-2\cdot \mathrm {sen} \,\left({\frac {x+y}{2}}\right)\mathrm {sen} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f2622fbcaccdb724cd8dbe3288be8dd8107b8b)
Partindo das fórmulas do seno da soma de arcos:
![{\displaystyle \mathrm {sen} \,(a+b)=\mathrm {sen} \,(a)\mathrm {cos} \,(b)+\mathrm {sen} \,(b)\mathrm {cos} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c760d7ade2bbe3109d398b7639897cbe15f8cfae)
![{\displaystyle \mathrm {sen} \,(a-b)=\mathrm {sen} \,(a)\mathrm {cos} \,(b)-\mathrm {sen} \,(b)\mathrm {cos} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d69b645a4c8a829fc56ddfad7d55962235fc96fb)
Somando-as membro a membro:
![{\displaystyle \mathrm {sen} \,(a+b)+\mathrm {sen} \,(a-b)=2\cdot \mathrm {sen} \,(a)\mathrm {cos} \,(b)(I)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5555fc3bb39c946742b4d02f9ebb7e7a97e84d)
Fazendo:
![{\displaystyle y=a-b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f56ef19dac70f6ca4fc48afeba84a9eb1913c04a)
Temos:
![{\displaystyle a={\frac {x+y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d963d371cc9c0a3a2c8cb592806843ddb098a21)
![{\displaystyle b={\frac {x-y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56945696ec2599f6ed8a5064ef73fee1afa3795d)
Substituindo a e b, em (I):
![{\displaystyle \mathrm {sen} \,(x)+\mathrm {sen} \,(y)=2\cdot \mathrm {sen} \,\left({\frac {x+y}{2}}\right)\mathrm {cos} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/771ff37894b920b252c93df056d0b2940b8758f3)
Procedendo da mesma forma, novamente a partir de:
![{\displaystyle \mathrm {sen} \,(a+b)=\mathrm {sen} \,(a)\mathrm {cos} \,(b)+\mathrm {sen} \,(b)\mathrm {cos} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c760d7ade2bbe3109d398b7639897cbe15f8cfae)
![{\displaystyle \mathrm {sen} \,(a-b)=\mathrm {sen} \,(a)\mathrm {cos} \,(b)-\mathrm {sen} \,(b)\mathrm {cos} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d69b645a4c8a829fc56ddfad7d55962235fc96fb)
Subtraindo-as membro a membro:
(II)
Substituindo a e b, em (II):
![{\displaystyle \mathrm {sen} \,(x)-\mathrm {sen} \,(y)=2\cdot \mathrm {sen} \,\left({\frac {x-y}{2}}\right)\mathrm {cos} \,\left({\frac {x+y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83150a9cb4cefdc3d6281343f2c750c5f564661)
Agora para a função cosseno
![{\displaystyle \mathrm {cos} \,(a+b)=\mathrm {cos} \,(a)\mathrm {cos} \,(b)-\mathrm {sen} \,(b)\mathrm {sen} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4e56efbf706d94be3159a73557d80bb6778bf4)
![{\displaystyle \mathrm {cos} \,(a-b)=\mathrm {cos} \,(a)\mathrm {cos} \,(b)+\mathrm {sen} \,(b)\mathrm {sen} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b9c1b5a48bdd357ffd4673192aa40d6ca589ae)
Somando-as membro a membro:
(III)
Substituindo a e b, em (III):
![{\displaystyle \mathrm {cos} \,(x)+\mathrm {cos} \,(y)=2\cdot \mathrm {cos} \,\left({\frac {x+y}{2}}\right)\mathrm {cos} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b976721df079c112155b51c4209de3c8ca1efd45)
E por fim:
![{\displaystyle \mathrm {cos} \,(a+b)=\mathrm {cos} \,(a)\mathrm {cos} \,(b)-\mathrm {sen} \,(b)\mathrm {sen} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4e56efbf706d94be3159a73557d80bb6778bf4)
![{\displaystyle \mathrm {cos} \,(a-b)=\mathrm {cos} \,(a)\mathrm {cos} \,(b)+\mathrm {sen} \,(b)\mathrm {sen} \,(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b9c1b5a48bdd357ffd4673192aa40d6ca589ae)
Subtraindo-as membro a membro:
(IV)
Substituindo a e b, em (IV):
![{\displaystyle \mathrm {cos} \,(x)-\mathrm {cos} \,(y)=-2\cdot \mathrm {sen} \,\left({\frac {x+y}{2}}\right)\mathrm {sen} \,\left({\frac {x-y}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f2622fbcaccdb724cd8dbe3288be8dd8107b8b)
Referências