Упростите:[latex] \sqrt{1+cos2 \alpha/2 } +cos \alpha [/latex][latex]cos \pi /5+cos2 \pi /5+cos4 \pi /5+cos6 \pi /5[/latex]Вычислите:cos36*cos72sin18*cos54[latex] cos2 \pi /7+cos4 \pi /7+cos6 \pi /7[/latex]

[latex]1)\sqrt{\frac{1+cos2a}{2}}+cosa=\sqrt{\frac{1+2cos^2a-1}{2}}+cosa=|cosa|+cosa=2cosa\\ cosa>0\\ 2)cos36+cos72+cos144+cos216\\ 1)cos36+cos216=2cos(144)*cos90=0\\ 2)cos72+cos144=2cos(108)*cos(36)=2cos(180-72)*cos36=-2cos72*cos36=\frac{-2cos72*cos36*sin36}{sin36}=\frac{-cos72*sin72}{sin36}=\frac{-sin144}{2sin36}=\frac{-sin36}{2sin36}=-\frac{1}{2}[/latex] [latex]3)cos36*cos72=\frac{cos36*cos72*2sin36}{2sin36}=\frac{sin72*cos72}{2sin36}=\frac{sin144}{4sin36}=\frac{sin36}{4sin36}=\frac{1}{4}\\ 4)sin18*cos54=sin18*cos(90-36)=sin18*sin36=\frac{2cos18*sin18*sin36}{2cos18}=\frac{sin36*sin36}{2cos18}[/latex] [latex]cos(\frac{2\pi}{7})+cos(\frac{4\pi}{7})+cos(\frac{6\pi}{7})\\ \frac{sin(\frac{\pi}{7})cos(\frac{2\pi}{7})+sin(\frac{\pi}{7})cos(\frac{4\pi}{7})+sin(\frac{\pi}{7})cos(\frac{6\pi}{7})}{sin(\frac{\pi}{7}})=-\frac{1}{2}[/latex]