已知sinb/sina=cos(a+b)求证:tanb=sin2a/(3-cos2a)a,b为锐角

问题描述:

已知sinb/sina=cos(a+b)求证:tanb=sin2a/(3-cos2a)
a,b为锐角

sinb/sina=cos(a+b)
sinb/sina=cosacosb - sinasinb
sinb = sinacosacosb - sin²a sinb
sinb/cosb = sinacosa - sin²asinb/cosb
tanb = (1/2)sin2a - sin²atana
(1+sin²a) tanb = (1/2)sin2a
tanb = sin2a /[2(1+sin²a)]
tanb = sin2a /[2+2sin²a]
= sin2a /[3-1+2sin²a]
= sin2a /(3-cos2a)