已知$alpha,eta,gamma$是三个互不相等的锐角,若$tanalpha=dfrac{sinetasingamma}{coseta-cosgamma}$则$ aneta=$______(表示成$alpha,gamma$)
解答:
$ an^2alpha+1=dfrac{sin^2etasin^2gamma}{(coseta-cosgamma)^2}+1$
$=dfrac{(1-cos^2eta)(1-cos^2gamma)+(coseta-cosgamma)^2}{(coseta-cosgamma)^2}$
$=dfrac{(1-cosetacosgamma)^2}{(coseta-cosgamma)^2}$
故$cosalpha=dfrac{coseta-cosgamma}{1-cosetacosgamma}$
解得$coseta=dfrac{cosalpha+cosgamma}{1+cosalphacosgamma}=dfrac{cosalpha-cos(pi-gamma)}{1-cosalphacos(pi-gamma)}$
考虑到$alpha$与$eta$的对称性与$gamma$与$pi-gamma$的对称性；
故$taneta=dfrac{sinalphasin(pi-gamma)}{cosalpha-cos(pi-gamma)}=dfrac{sinalphasingamma}{cosalpha+cosgamma}$