Question

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

Answer 1

$\dfrac{\sin(2\alpha)}{\sin(2\alpha)+\cos(2\alpha)}=\dfrac{2\sin\alpha\cos\alpha}{2\sin\alpha\cos\alpha+\cos^2\alpha-\sin^2\alpha}=\medskip\\\dfrac{\cos^2\alpha\cdot2\mathrm{tg}~\alpha}{\cos^2\alpha\left(2\mathrm{tg}~\alpha+1-\mathrm{tg}^2\alpha\right)}=\dfrac{2\mathrm{tg}~\alpha}{2\mathrm{tg}~\alpha-\mathrm{tg}^2\alpha+1}$

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

Ответ. $\dfrac{4}{7}$