![\cos \alpha=\cos^2\frac{\alpha}{2}-\sin^2\frac{\alpha}{2}=(\cos\frac{\alpha}{2}-\sin\frac{\alpha}{2})(\cos\frac{\alpha}{2}+\sin\frac{\alpha}{2})=\\ \\ =\cos\frac{\alpha}{2}(1-{\rm tg}\frac{\alpha}{2})\cdot \cos\frac{\alpha}{2}(1+{\rm tg}\frac{\alpha}{2})=\cos^2\frac{\alpha}{2}(1-{\rm tg}^2\frac{\alpha}{2})=\boxed{\dfrac{1-{\rm tg}^2\frac{\alpha}{2}}{1+{\rm tg}^2\frac{\alpha}{2}}}](https://tex.z-dn.net/?f=%5Ccos+%5Calpha%3D%5Ccos%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D-%5Csin%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D%3D%28%5Ccos%5Cfrac%7B%5Calpha%7D%7B2%7D-%5Csin%5Cfrac%7B%5Calpha%7D%7B2%7D%29%28%5Ccos%5Cfrac%7B%5Calpha%7D%7B2%7D%2B%5Csin%5Cfrac%7B%5Calpha%7D%7B2%7D%29%3D%5C%5C+%5C%5C+%3D%5Ccos%5Cfrac%7B%5Calpha%7D%7B2%7D%281-%7B%5Crm+tg%7D%5Cfrac%7B%5Calpha%7D%7B2%7D%29%5Ccdot+%5Ccos%5Cfrac%7B%5Calpha%7D%7B2%7D%281%2B%7B%5Crm+tg%7D%5Cfrac%7B%5Calpha%7D%7B2%7D%29%3D%5Ccos%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D%281-%7B%5Crm+tg%7D%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D%29%3D%5Cboxed%7B%5Cdfrac%7B1-%7B%5Crm+tg%7D%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%7B1%2B%7B%5Crm+tg%7D%5E2%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%7D)
P.S.
- известное тождество.
2) Так как α ∈ (0°;90°) - I четверть, то в этой четверти синус и косинус положительные, тогда из тождества
, найдем косинус
![\cos \alpha =\sqrt{\dfrac{1}{1+{\rm tg}^2\alpha}}=\sqrt{\dfrac{1}{1+\bigg(\dfrac{2}{3}\bigg)^2}}=\sqrt{\dfrac{9}{9+4}}=\dfrac{3}{\sqrt{13}}](https://tex.z-dn.net/?f=%5Ccos+%5Calpha+%3D%5Csqrt%7B%5Cdfrac%7B1%7D%7B1%2B%7B%5Crm+tg%7D%5E2%5Calpha%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B1%7D%7B1%2B%5Cbigg%28%5Cdfrac%7B2%7D%7B3%7D%5Cbigg%29%5E2%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B9%7D%7B9%2B4%7D%7D%3D%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D)
![\sin\alpha=\sqrt{1-\cos^2\alpha}=\sqrt{1-\bigg(\dfrac{3}{\sqrt{13}}\bigg)^2}=\dfrac{2}{\sqrt{13}}](https://tex.z-dn.net/?f=%5Csin%5Calpha%3D%5Csqrt%7B1-%5Ccos%5E2%5Calpha%7D%3D%5Csqrt%7B1-%5Cbigg%28%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cbigg%29%5E2%7D%3D%5Cdfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D)
![\sin2\alpha=2\sin\alpha\cos\alpha=2\cdot\dfrac{2}{\sqrt{13}}\cdot \dfrac{3}{\sqrt{13}}=\dfrac{12}{13}\\ \\ \cos2\alpha=\cos^2\alpha-\sin^2\alpha=\bigg(\dfrac{3}{\sqrt{13}}\bigg)^2-\bigg(\dfrac{2}{\sqrt{13}}\bigg)^2=\dfrac{5}{13}\\ \\ \\{\rm tg}2\alpha=\dfrac{\sin2\alpha}{\cos2\alpha}=\dfrac{\dfrac{12}{13}}{\dfrac{5}{13}}=\dfrac{12}{5}=2.4](https://tex.z-dn.net/?f=%5Csin2%5Calpha%3D2%5Csin%5Calpha%5Ccos%5Calpha%3D2%5Ccdot%5Cdfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%5Ccdot+%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%3D%5Cdfrac%7B12%7D%7B13%7D%5C%5C+%5C%5C+%5Ccos2%5Calpha%3D%5Ccos%5E2%5Calpha-%5Csin%5E2%5Calpha%3D%5Cbigg%28%5Cdfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%5Cbigg%29%5E2-%5Cbigg%28%5Cdfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%5Cbigg%29%5E2%3D%5Cdfrac%7B5%7D%7B13%7D%5C%5C+%5C%5C+%5C%5C%7B%5Crm+tg%7D2%5Calpha%3D%5Cdfrac%7B%5Csin2%5Calpha%7D%7B%5Ccos2%5Calpha%7D%3D%5Cdfrac%7B%5Cdfrac%7B12%7D%7B13%7D%7D%7B%5Cdfrac%7B5%7D%7B13%7D%7D%3D%5Cdfrac%7B12%7D%7B5%7D%3D2.4)
y=x^2-4/x-2 , дробь имеет смысл, когда знаменатель не равен нулю,числитель любой
Решение смотри в приложении