Question

Помогите решить sin2α/sin2α+cos2α, если tgα=-2

Answer 1

$dfrac{sin(2alpha)}{sin(2alpha)+cos(2alpha)}=dfrac{2sinalphacosalpha}{2sinalphacosalpha+cos^2alpha-sin^2alpha}=medskip\dfrac{cos^2alphacdot2mathrm{tg}~alpha}{cos^2alphaleft(2mathrm{tg}~alpha+1-mathrm{tg}^2alpharight)}=dfrac{2mathrm{tg}~alpha}{2mathrm{tg}~alpha-mathrm{tg}^2alpha+1}$

$mathrm{tg}~alpha=-2Rightarrow dfrac{sin(2alpha)}{sin(2alpha)+cos(2alpha)}=dfrac{2mathrm{tg}~alpha}{2mathrm{tg}~alpha-mathrm{tg}^2alpha+1}=dfrac{2(-2)}{2(-2)-(-2)^2+1}=medskip\=dfrac{-4}{-4-4+1}=dfrac{-4}{-7}=dfrac{4}{7}$

Ответ. $dfrac{4}{7}$