<span>а ) а^16*а^4=a^16+4=a^20
б)а^16:а^4 =a^16-4=a^12
в)(а^16)^4 =a^16*4=a^64</span>
1.
а)
![cos2x+tg^2x*cos2x-1=cos2x(1+tg^2x)-1= \\ \\ =cos2x* \frac{1}{cos^2x}-1= \frac{cos^2x-sin^2x}{cos^2x}-1= \\ \\ = \frac{cos^2x}{cos^2x}- \frac{sin^2x}{cos^2x}-1=1-tg^2x-1=-tg^2x \\ \\ -tg^2x=-tg^2x](https://tex.z-dn.net/?f=cos2x%2Btg%5E2x%2Acos2x-1%3Dcos2x%281%2Btg%5E2x%29-1%3D+%5C%5C++%5C%5C+%0A%3Dcos2x%2A+%5Cfrac%7B1%7D%7Bcos%5E2x%7D-1%3D+%5Cfrac%7Bcos%5E2x-sin%5E2x%7D%7Bcos%5E2x%7D-1%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7Bcos%5E2x%7D%7Bcos%5E2x%7D-+%5Cfrac%7Bsin%5E2x%7D%7Bcos%5E2x%7D-1%3D1-tg%5E2x-1%3D-tg%5E2x+%5C%5C++%5C%5C+%0A-tg%5E2x%3D-tg%5E2x++++)
Что и требовалось доказать.
б)
![(sin4x-sin5x)-(sin6x-sin7x)= \\ \\ =2sin \frac{4x-5x}{2}cos \frac{4x+5x}{2}-2sin \frac{6x-7x}{2}cos \frac{6x+7x}{2}= \\ \\ =2sin(- \frac{x}{2} )cos4.5x-2sin(- \frac{x}{2} )cos6.5x= \\ \\ =2sin(- \frac{x}{2} )(cos4.5x-cos6.5x)= \\ \\ =-2sin( \frac{x}{2} )(-2sin \frac{4.5x+6.5x}{2}sin \frac{4.5x-6.5x}{2} )= \\ \\ =-2sin( \frac{x}{2} )(-2sin \frac{11x}{2} sin(- \frac{2x}{2} ))= \\ \\ =-4sin \frac{x}{2}sinxsin \frac{11x}{2}](https://tex.z-dn.net/?f=%28sin4x-sin5x%29-%28sin6x-sin7x%29%3D+%5C%5C++%5C%5C+%0A%3D2sin+%5Cfrac%7B4x-5x%7D%7B2%7Dcos+%5Cfrac%7B4x%2B5x%7D%7B2%7D-2sin+%5Cfrac%7B6x-7x%7D%7B2%7Dcos+%5Cfrac%7B6x%2B7x%7D%7B2%7D%3D+%5C%5C++%5C%5C+%0A%3D2sin%28-+%5Cfrac%7Bx%7D%7B2%7D+%29cos4.5x-2sin%28-+%5Cfrac%7Bx%7D%7B2%7D+%29cos6.5x%3D+%5C%5C++%5C%5C+%0A%3D2sin%28-+%5Cfrac%7Bx%7D%7B2%7D+%29%28cos4.5x-cos6.5x%29%3D+%5C%5C++%5C%5C+%0A%3D-2sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29%28-2sin+%5Cfrac%7B4.5x%2B6.5x%7D%7B2%7Dsin+%5Cfrac%7B4.5x-6.5x%7D%7B2%7D++%29%3D+%5C%5C++%5C%5C+%0A%3D-2sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29%28-2sin+%5Cfrac%7B11x%7D%7B2%7D+sin%28-+%5Cfrac%7B2x%7D%7B2%7D+%29%29%3D+%5C%5C++%5C%5C+%0A%3D-4sin+%5Cfrac%7Bx%7D%7B2%7Dsinxsin+%5Cfrac%7B11x%7D%7B2%7D++++++)
Что и требовалось доказать.
2.
![tg( \frac{x}{3}+ \frac{ \pi }{4} )+tg( \frac{x}{3}- \frac{ \pi }{4} )= \\ \\ = \frac{tg \frac{x}{3} +tg \frac{ \pi }{4} }{1-tg \frac{x}{3}tg \frac{ \pi }{4} }+ \frac{tg \frac{x}{3}-tg \frac{ \pi }{4} }{1+tg \frac{x}{3}tg \frac{ \pi }{4} }= \\ \\ = \frac{tg \frac{x}{3}+1 }{1-tg \frac{x}{3} }+ \frac{tg \frac{x}{3}-1 }{1+tg \frac{x}{3} }= \\ \\ = \frac{(tg \frac{x}{3}+1 )^2+(tg \frac{x}{3} -1)(1-tg \frac{x}{3} )}{(1-tg \frac{x}{3} )(1+tg \frac{x}{3} )}=](https://tex.z-dn.net/?f=tg%28+%5Cfrac%7Bx%7D%7B3%7D%2B+%5Cfrac%7B+%5Cpi+%7D%7B4%7D++%29%2Btg%28+%5Cfrac%7Bx%7D%7B3%7D-+%5Cfrac%7B+%5Cpi+%7D%7B4%7D++%29%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7Btg+%5Cfrac%7Bx%7D%7B3%7D+%2Btg+%5Cfrac%7B+%5Cpi+%7D%7B4%7D+%7D%7B1-tg+%5Cfrac%7Bx%7D%7B3%7Dtg+%5Cfrac%7B+%5Cpi+%7D%7B4%7D++%7D%2B+%5Cfrac%7Btg+%5Cfrac%7Bx%7D%7B3%7D-tg+%5Cfrac%7B+%5Cpi+%7D%7B4%7D++%7D%7B1%2Btg+%5Cfrac%7Bx%7D%7B3%7Dtg+%5Cfrac%7B+%5Cpi+%7D%7B4%7D++%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7Btg+%5Cfrac%7Bx%7D%7B3%7D%2B1+%7D%7B1-tg+%5Cfrac%7Bx%7D%7B3%7D+%7D%2B+%5Cfrac%7Btg+%5Cfrac%7Bx%7D%7B3%7D-1+%7D%7B1%2Btg+%5Cfrac%7Bx%7D%7B3%7D+%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B%28tg+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2%2B%28tg+%5Cfrac%7Bx%7D%7B3%7D+-1%29%281-tg+%5Cfrac%7Bx%7D%7B3%7D+%29%7D%7B%281-tg+%5Cfrac%7Bx%7D%7B3%7D+%29%281%2Btg+%5Cfrac%7Bx%7D%7B3%7D+%29%7D%3D+++++)
![= \frac{(tg \frac{x}{3}+1 )^2-(tg \frac{x}{3}-1 )^2}{1-tg^2 \frac{x}{3} }= \frac{(tg \frac{x}{3}+1-tg \frac{x}{3}+1 )(tg \frac{x}{3}+1+tg \frac{x}{3}-1 )}{1-tg^2 \frac{x}{3} }= \\ \\ = \frac{2*2tg \frac{x}{3} }{1-tg^2 \frac{x}{3} }=2* \frac{2tg \frac{x}{3} }{1-tg^2 \frac{x}{3} }=2tg(2* \frac{x}{3} ) =2tg \frac{2x}{3}](https://tex.z-dn.net/?f=%3D+%5Cfrac%7B%28tg+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2-%28tg+%5Cfrac%7Bx%7D%7B3%7D-1+%29%5E2%7D%7B1-tg%5E2+%5Cfrac%7Bx%7D%7B3%7D+%7D%3D+%5Cfrac%7B%28tg+%5Cfrac%7Bx%7D%7B3%7D%2B1-tg+%5Cfrac%7Bx%7D%7B3%7D%2B1++%29%28tg+%5Cfrac%7Bx%7D%7B3%7D%2B1%2Btg+%5Cfrac%7Bx%7D%7B3%7D-1++%29%7D%7B1-tg%5E2+%5Cfrac%7Bx%7D%7B3%7D+%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B2%2A2tg+%5Cfrac%7Bx%7D%7B3%7D+%7D%7B1-tg%5E2+%5Cfrac%7Bx%7D%7B3%7D+%7D%3D2%2A+%5Cfrac%7B2tg+%5Cfrac%7Bx%7D%7B3%7D+%7D%7B1-tg%5E2+%5Cfrac%7Bx%7D%7B3%7D+%7D%3D2tg%282%2A+%5Cfrac%7Bx%7D%7B3%7D+%29+%3D2tg+%5Cfrac%7B2x%7D%7B3%7D++++)
3.
2cos3x cos4x - cos7x=2cos3x cos4x - cos(3x+4x)=
=2cos3x cos4x -(cos3x cos4x - sin3x sin4x)=
=2cos3x cos4x - cos3x cos4x + sin3x sin4x =
= cos3x cos4x + sin3x sin4x= cos(3x-4x)=cos(-x)=cosx=
=cos(2*(x/2))=cos²(x/2)-sin²(x/2)=cos²(x/2)-(1-cos²(x/2))=
=cos²(x/2)-1+cos²(x/2)=2cos²(x/2)-1
2*(√0.8)² -1= 2*0.8 -1= 1.6-1=0.6
4.
Угол х находится в 3-ей четверти.
Знак в 3-ей четверти sinx - "-"; cosx - "-"; tgx - "+".
cos(π/2+x)= -sinx
-sinx = 12/13
sinx= - 12/13
![cosx= - \sqrt{1-sin^2x}=- \sqrt{1-(- \frac{12}{13} )^2}=- \sqrt{1- \frac{144}{169} }= \\ \\ =- \sqrt{ \frac{25}{169} }=- \frac{5}{13} \\ \\ tgx= \frac{sinx}{cosx}= \frac{-12}{13}:(- \frac{5}{13} )= \frac{12}{5}](https://tex.z-dn.net/?f=cosx%3D+-+%5Csqrt%7B1-sin%5E2x%7D%3D-+%5Csqrt%7B1-%28-+%5Cfrac%7B12%7D%7B13%7D+%29%5E2%7D%3D-+%5Csqrt%7B1-+%5Cfrac%7B144%7D%7B169%7D+%7D%3D+%5C%5C++%5C%5C+%0A%3D-+%5Csqrt%7B+%5Cfrac%7B25%7D%7B169%7D+%7D%3D-+%5Cfrac%7B5%7D%7B13%7D+%5C%5C++%5C%5C+%0Atgx%3D+%5Cfrac%7Bsinx%7D%7Bcosx%7D%3D+%5Cfrac%7B-12%7D%7B13%7D%3A%28-+%5Cfrac%7B5%7D%7B13%7D+%29%3D+%5Cfrac%7B12%7D%7B5%7D++++++++)
0,9x-0,6x+1,8=0,4x-2,6
0,3x+1,8=0,4x-2,6
0,3x-0,4x=-2,6-1,8
-0,1x=-4,4
x=44
2) -1,2x+0,4+6,4x-2,4=5-3,8x-4
5,2x-2=1-3,8x
5,2x+3,8x=1+2
9x=3
x=1/3
3) 4/7*(14/25-21/5y)+2/5=5/13*(13/25-13/2y)
8/25-12/5y+2/5=1/5-5/2y
18/25-12/5y=1/5-5/2y /*50
36-120y=10-125y
-120y+125y=10-36
5y=-26
y=-26/5=-5,2