Question

468(1) 469(1,3) )))))))

Answer 1

$1)qquad cos2a=?\\ frac{cosa-2sina}{sina-2cosa} =-0,5; ; ; Rightarrow \\ frac{cosa-2sina}{sina-2cosa} =Big [ frac{:cosa}{:cosa} Big ]= frac{1-2tga}{tga-2}=-frac{1}{2}; ; ; Rightarrow \\2cdot ( 1-2tga)=-(tga-2)\\2-2tga=-tga+2\\tga=0\\Formyla:quad cosx=frac{1-tg^2frac{x}{2}}{1+tg^2frac{x}{2}}; ; ; Rightarrow \\cos2a=frac{1-tg^2a}{1+tg^2a}= frac{1-0}{1+0} =1$

$2)quad 8cdot sin^2frac{15pi }{6}cdot cos^2frac{17pi }{6}=8sin^2(2pi +frac{3pi }{6})cdot cos^2(3pi -frac{pi }{6})=\\=8cdot sin^2frac{pi}{2}cdot cos^2frac{pi}{6}=8cdot 1^2cdot (frac{sqrt3}{2})^2=8cdot frac{3}{4}=6\\3)quad tgfrac{7pi}{8}+ctgfrac{7pi}{8}=tg(pi -frac{pi}8})+ctg(pi -frac{pi}{8})=$

$=-tgfrac{pi}{8}-ctgfrac{pi}{8}=-frac{sinfrac{pi}{8}}{cosfrac{pi}{8}}-frac{cosfrac{pi}{8}}{sinfrac{pi}{8}}=- frac{sin^2frac{pi}{8}+cos^2frac{pi}{8}}{sinfrac{pi}{8}cdot cosfrac{pi }{8}}=$

$=-frac{1}{frac{1}{2}sinfrac{pi}{4}}=-frac{2}{frac{sqrt2}{2}}=-frac{2cdot 2}{sqrt2}=-2sqrt2$