Question

Mostre que, se sen x + cos x = m, então sen 2x = m² - 1:

Answer 1

$Ol\acute{a}~~~\mathbb{LETIXA} \\ \\ Seja~: \\ \\ senx+cosx=m~~---\ \textgreater \ elevamos~ao~quadrado~a~igualdade \\ \\ (senx+cosx)^{2} =m\²~ \\ \\ senx^{2} +2senx.cosx+cosx^{2}=m\² \\ \\ \underbrace{senx^{2}+ cosx^{2}}_{1} +2senx.cosx=m\² \\ \\ ~~~~~~~~~1+2senx.cosx=m\² \\ \\~~~~~~~~~~~~~\boxed{ 2senx.cosx=m\²-1}~~-----\ \textgreater \ (I)$


$\mathbb{uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu} \\ Por~identidade~sabemos~ que: \\ sen(x+x)=senx.cosx+cosx.senx \\ \\ sen(2x)=\underbrace{2senx.consx}_{ m^{2}-1 }~~~---\ \textgreater \ substituindo~(I)~temos: \\ \\ \\ \boxed{\boxed{sen(2x)= m^{2}-1}}~---\ \textgreater \ queda~demostrado \\ \\ \mathbb{wwwwwwwwwwwwwwwwwwwwwvvvvvvvvvvvvvvvvvv} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Espero~ter~ajudado!!$