(a)
![\frac{4}{3}tg( \pi -arcsin(_\frac{3}{5}) =\frac{4}{3}tg( \pi +arcsin(\frac{3}{5} ) \frac{4}{3}tg( \pi -arcsin(_\frac{3}{5}) =\frac{4}{3}tg( \pi +arcsin(\frac{3}{5} )](https://tex.z-dn.net/?f=+%5Cfrac%7B4%7D%7B3%7Dtg%28+%5Cpi+-arcsin%28_%5Cfrac%7B3%7D%7B5%7D%29+%3D%5Cfrac%7B4%7D%7B3%7Dtg%28+%5Cpi+%2Barcsin%28%5Cfrac%7B3%7D%7B5%7D+%29+)
![arcsin( \frac{3}{5})=x arcsin( \frac{3}{5})=x](https://tex.z-dn.net/?f=+arcsin%28+%5Cfrac%7B3%7D%7B5%7D%29%3Dx+)
![sinx=\frac{3}{5} sinx=\frac{3}{5}](https://tex.z-dn.net/?f=+++++++sinx%3D%5Cfrac%7B3%7D%7B5%7D+)
![\frac{4}{3} tg( \pi +arcsin(\frac{3}{5} )=\frac{4}{3} \frac{tg(\pi)+tg(x)}{1-tg( \pi )tg(x)}=\frac{4}{3} \frac{tg(x)}{1} \frac{4}{3} tg( \pi +arcsin(\frac{3}{5} )=\frac{4}{3} \frac{tg(\pi)+tg(x)}{1-tg( \pi )tg(x)}=\frac{4}{3} \frac{tg(x)}{1}](https://tex.z-dn.net/?f=+%5Cfrac%7B4%7D%7B3%7D+tg%28+%5Cpi+%2Barcsin%28%5Cfrac%7B3%7D%7B5%7D+%29%3D%5Cfrac%7B4%7D%7B3%7D+%5Cfrac%7Btg%28%5Cpi%29%2Btg%28x%29%7D%7B1-tg%28+%5Cpi+%29tg%28x%29%7D%3D%5Cfrac%7B4%7D%7B3%7D+%5Cfrac%7Btg%28x%29%7D%7B1%7D+)
Находим значение tg
![tg(x)=\frac{sinx}{cosx} tg(x)=\frac{sinx}{cosx}](https://tex.z-dn.net/?f=+tg%28x%29%3D%5Cfrac%7Bsinx%7D%7Bcosx%7D+)
sin²x+cos²x=1- sin²x=9/25
![cosx= \sqrt{1-sin^{2}x}=\frac{4}{5} cosx= \sqrt{1-sin^{2}x}=\frac{4}{5}](https://tex.z-dn.net/?f=+cosx%3D+%5Csqrt%7B1-sin%5E%7B2%7Dx%7D%3D%5Cfrac%7B4%7D%7B5%7D+)
![tg(x)=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4} tg(x)=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}](https://tex.z-dn.net/?f=+tg%28x%29%3D%5Cfrac%7B%5Cfrac%7B3%7D%7B5%7D%7D%7B%5Cfrac%7B4%7D%7B5%7D%7D%3D%5Cfrac%7B3%7D%7B4%7D+)
Подставляем в решение
![\frac{4}{3}*\frac{3}{4}=1 \frac{4}{3}*\frac{3}{4}=1](https://tex.z-dn.net/?f=+%5Cfrac%7B4%7D%7B3%7D%2A%5Cfrac%7B3%7D%7B4%7D%3D1+)
b)
![log_{\sqrt{2}}(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=2 log_{\sqrt{2}}(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=2](https://tex.z-dn.net/?f=+log_%7B%5Csqrt%7B2%7D%7D%282%5E%7B%5Cfrac%7Bx-2%7D%7Bx%7D%7D-5%2A2%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%2B4%29%3D2++)
0 " alt="(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)>0 " align="absmiddle" class="latex-formula">
![(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=0 (2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=0](https://tex.z-dn.net/?f=%282%5E%7B%5Cfrac%7Bx-2%7D%7Bx%7D%7D-5%2A2%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%2B4%29%3D0+)
![(2^{1-\frac{2}{x}}-5*2^{\frac{1}{x}}+4)=0 (2^{1-\frac{2}{x}}-5*2^{\frac{1}{x}}+4)=0](https://tex.z-dn.net/?f=%282%5E%7B1-%5Cfrac%7B2%7D%7Bx%7D%7D-5%2A2%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%2B4%29%3D0+)
![2^{-\frac{1}{x}}=y 2^{-\frac{1}{x}}=y](https://tex.z-dn.net/?f=+2%5E%7B-%5Cfrac%7B1%7D%7Bx%7D%7D%3Dy+)
![2*y^{2}-5y+4=0 2*y^{2}-5y+4=0](https://tex.z-dn.net/?f=+2%2Ay%5E%7B2%7D-5y%2B4%3D0+)
D=25-4*4*2<0. Ветви параболы направлены вверх=> y∈R
=>x∈(-∞:0)(0;+∞)
![(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=2 (2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+4)=2](https://tex.z-dn.net/?f=+%282%5E%7B%5Cfrac%7Bx-2%7D%7Bx%7D%7D-5%2A2%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%2B4%29%3D2)
![(2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+2)=0 (2^{\frac{x-2}{x}}-5*2^{\frac{1}{x}}+2)=0](https://tex.z-dn.net/?f=+%282%5E%7B%5Cfrac%7Bx-2%7D%7Bx%7D%7D-5%2A2%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%2B2%29%3D0)
![2^{-\frac{1}{x}}=y 2^{-\frac{1}{x}}=y](https://tex.z-dn.net/?f=+2%5E%7B-%5Cfrac%7B1%7D%7Bx%7D%7D%3Dy+)
![2*y^{2}-5y+2=0 2*y^{2}-5y+2=0](https://tex.z-dn.net/?f=+2%2Ay%5E%7B2%7D-5y%2B2%3D0+)
D= 25-16=9=3²
![y1=\frac{5+3} {4}=2 y1=\frac{5+3} {4}=2](https://tex.z-dn.net/?f=+y1%3D%5Cfrac%7B5%2B3%7D+%7B4%7D%3D2+)
![y2=\frac{5-3} {4}=\frac{1}{2} y2=\frac{5-3} {4}=\frac{1}{2}](https://tex.z-dn.net/?f=+y2%3D%5Cfrac%7B5-3%7D+%7B4%7D%3D%5Cfrac%7B1%7D%7B2%7D+)
, ![-\frac{1}{x}=\frac{1}{2} -\frac{1}{x}=\frac{1}{2}](https://tex.z-dn.net/?f=+-%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7B1%7D%7B2%7D+)
, ![x2=-2 x2=-2](https://tex.z-dn.net/?f=+x2%3D-2+)