Question

426(1,3)
427(1,3)
428(1,3)
429(1,3,5)

Answer 1

$426.; ; 1); ctgx+frac{sinx}{1+cosx}= frac{cosx}{sinx} + frac{2sinfrac{x}{2}cdot cosfrac{x}{2}}{2cos^2frac{x}{2}} = \\=frac{2cos^2frac{x}{2}-1}{2sinfrac{x}{2}cdot cosfrac{x}{2}} +frac{sinfrac{x}{2}}{cosfrac{x}{2}}= frac{2cos^2frac{x}{2}-1+2sin^2frac{x}{2}}{2sinfrac{x}{2}cdot cosfrac{x}{2}} =frac{2-1}{sinx}=frac{1}{sinx} \\3); ; frac{1-sin^2x}{1-cos^2x}=frac{cos^2x}{sin^2x} =ctg^2x=frac{1}{tg^2x}$

$427.; ; 1); y=1-(cos^2x-sin^2x)=1-cos2x\\-1 leq cos2x leq 1; ; ; ; to ; ; ; ; -1 leq -cos2x leq 1\\-1+1 leq 1-cos2x leq 1+1\\0 leq 1-cos2x leq 2; ; to ; ; ; y_{naibol.}=2\\2); ; y=cos^2acdot tg^2a+5cos^2a-1=sin^2a+5cos^2a-1=\\=(1-cos^2a)+5cos^2a-1=4cos^2a\\-1 leq cosa leq 1\\0 leq cos^2a leq 1; ; ; to ; ; ; 0 leq 4cos^2a leq 4\\y_{naibol.}=4$

$428.; ; 1); 1+sin frac{pi}{6}+sin^2 frac{pi}{6}+sin^3frac{pi}{6} =1+ frac{1}{2}+frac{1}{4} +frac{1}{8}=\\= frac{3}{2}+frac{3}{8}=frac{15}{8}\\3); ; 1-tg frac{pi}{6}+tg^2frac{pi}{6}-tg^3frac{pi}{6} =1-frac{1}{sqrt3}+frac{1}{3}-frac{1}{3sqrt3}=\\=frac{4}{3}-frac{4}{3sqrt3}= frac{4sqrt3-4}{3sqrt3} =frac{4(sqrt3-1)}{3sqrt3}= frac{4sqrt3(sqrt3-1)}{9}$

$429.; ; 1); frac{cosx}{1+sinx} +tgx= frac{cosx}{1+sinx} +frac{sinx}{cosx} =\\=frac{cos^2x+sinx+sin^2x}{1+sinx} = frac{1+sinx}{1+sinx} =1\\3); ; frac{1-sin^2x}{1-cos^2x} +tgxcdot ctgx= frac{cos^2x}{sin^2x} +1=ctg^2x+1=frac{1}{sin^2x}\\5); ; (ctga+tga)^2-(ctga-tga)^2=\\=(ctga+tga-(ctga-tga))cdot (ctga+tga+(ctga-tga))=\\=2tgacdot 2ctga=4tgacdot ctga=4cdot 1 =4$