![1+tg ^{2} \alpha = \frac{tg ^{2} \alpha }{Sin ^{2} \alpha }\\\\ \frac{tg ^{2} \alpha }{Sin ^{2} \alpha } = \frac{ \frac{Sin ^{2} \alpha }{Cos ^{2} \alpha } }{Sin ^{2} \alpha } = \frac{Sin ^{2} \alpha }{Cos ^{2} \alpha *Sin ^{2} \alpha }= \frac{1}{Cos ^{2} \alpha } =1+tg ^{2} \alpha \\\\1+tg ^{2} \alpha =1+tg ^{2} \alpha](https://tex.z-dn.net/?f=1%2Btg+%5E%7B2%7D+%5Calpha+%3D+%5Cfrac%7Btg+%5E%7B2%7D+%5Calpha++%7D%7BSin+%5E%7B2%7D+%5Calpha++%7D%5C%5C%5C%5C+%5Cfrac%7Btg+%5E%7B2%7D+%5Calpha++%7D%7BSin+%5E%7B2%7D+%5Calpha++%7D+%3D+%5Cfrac%7B+%5Cfrac%7BSin+%5E%7B2%7D+%5Calpha++%7D%7BCos+%5E%7B2%7D++%5Calpha+%7D+%7D%7BSin+%5E%7B2%7D++%5Calpha+%7D+%3D+%5Cfrac%7BSin+%5E%7B2%7D+%5Calpha++%7D%7BCos+%5E%7B2%7D+%5Calpha+%2ASin+%5E%7B2%7D++%5Calpha++%7D%3D+%5Cfrac%7B1%7D%7BCos+%5E%7B2%7D+%5Calpha++%7D+%3D1%2Btg+%5E%7B2%7D+%5Calpha+%5C%5C%5C%5C1%2Btg+%5E%7B2%7D++%5Calpha+%3D1%2Btg+%5E%7B2%7D+%5Calpha++++++)
Тождество доказано, при этом была использована формула:
![\frac{1}{Cos ^{2} \alpha }=1+tg ^{2} \alpha](https://tex.z-dn.net/?f=+%5Cfrac%7B1%7D%7BCos+%5E%7B2%7D+%5Calpha++%7D%3D1%2Btg+%5E%7B2%7D+%5Calpha+++)
![\frac{Cos \alpha -Cos ^{3} \alpha }{Sin ^{2} \alpha } =-Sin( \frac{3 \pi }{2} - \alpha )\\\\\\ \frac{Cos \alpha -Cos ^{3} \alpha }{Sin ^{2} \alpha }= \frac{Cos \alpha (1-Cos ^{2} \alpha ) }{Sin ^{2} \alpha } = \frac{Cos \alpha *Sin ^{2} \alpha }{Sin ^{2} \alpha } =Cos \alpha\\\\\\-Sin( \frac{3 \pi }{2} - \alpha ) =Cos \alpha \\\\\\Cos \alpha =Cos \alpha](https://tex.z-dn.net/?f=+%5Cfrac%7BCos+%5Calpha+-Cos+%5E%7B3%7D+%5Calpha++%7D%7BSin+%5E%7B2%7D+%5Calpha++%7D+%3D-Sin%28+%5Cfrac%7B3+%5Cpi+%7D%7B2%7D+-+%5Calpha+%29%5C%5C%5C%5C%5C%5C+%5Cfrac%7BCos+%5Calpha+-Cos+%5E%7B3%7D++%5Calpha+%7D%7BSin+%5E%7B2%7D++%5Calpha+%7D%3D+%5Cfrac%7BCos+%5Calpha+%281-Cos+%5E%7B2%7D+%5Calpha+%29+%7D%7BSin+%5E%7B2%7D++%5Calpha+%7D+%3D+%5Cfrac%7BCos+%5Calpha+%2ASin+%5E%7B2%7D+%5Calpha++%7D%7BSin+%5E%7B2%7D+%5Calpha++%7D+%3DCos+%5Calpha%5C%5C%5C%5C%5C%5C-Sin%28+%5Cfrac%7B3+%5Cpi+%7D%7B2%7D+-+%5Calpha+%29+%3DCos+%5Calpha+%5C%5C%5C%5C%5C%5CCos+%5Calpha+%3DCos+%5Calpha+++)
Тождество доказано
Разделим знаменатели на 2
(2х+7)/3=х/2 - пропорция
(2х+7)*2=3х
4х+14=3х
х=-14 - это ответ.
M/10=m-3/8
m=10*m-30/8
9m=30/8
m=5/12
Область определения - вся числовая ось, от минус бесконечности до плюс бесконечности.
пусть х больше y:
![\left \{ {{x^2-y^2=6} \atop {(x-2)^2-(y-2)^2=18}} \right.](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%5E2-y%5E2%3D6%7D+%5Catop+%7B%28x-2%29%5E2-%28y-2%29%5E2%3D18%7D%7D+%5Cright.)
![\left \{ {{x^2-y^2=6} \atop {x^2-4x+4-y^2+4y-4=18}} \right.](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%5E2-y%5E2%3D6%7D+%5Catop+%7Bx%5E2-4x%2B4-y%5E2%2B4y-4%3D18%7D%7D+%5Cright.)
Отнимем из второго первое уравнение:
4y-4x=12;
y=3+x;
Подставим в первое уравнение полученное только что:
![x^2-(3+x)^2=6;](https://tex.z-dn.net/?f=x%5E2-%283%2Bx%29%5E2%3D6%3B)
![x^2-9-6x-x^2=6;](https://tex.z-dn.net/?f=x%5E2-9-6x-x%5E2%3D6%3B)
x=-5/2=-2,5;
y=3-2,5=0,5;
x+y=0,5-2,5=-2;