Question

докажите 1+2cosx+cos2x/1+cos2x-2cosx=-ctg^2 x/2

Answer 1

$frac{1+2cosx+cos2x}{1+cos2x-2cosx}=frac{(1+cos2x)+2cosx}{(1+cos2x)-2cosx}=; [1+cos2 alpha =2cos^2 alpha ]; =\\=frac{2cos^2x+2cosx}{2cos^2x-2cosx}=frac{2cosx(cosx+1)}{2cosx(cosx-1)}=\\=[1+cos alpha =2cos^2frac{ alpha }{2}},; 1-cos alpha =2sin^2frac{ alpha }{2}]=\\=frac{2cos^2frac{x}{2}}{-2sin^2frac{x}{2}}=-ctg^2frac{x}{2}$