Question

sin^4x-cos^4x если tgx/2=0.5

Answer 1

$tgfrac{x}{2}=0,5\\sin^4x-cos^4x=(sin^2x-cos^2x)(sin^2x+cos^2x)=\\=sin^2x-cos^2x=(sinx-cosx)(sinx+cosx)=\\=left (frac{2tgfrac{x}{2}}{1+tg^2frac{x}{2}}- frac{1-tg^2frac{x}{2}}{1+tg^2frac{x}{2}} right )left ( frac{2tgfrac{x}{2}}{1+tg^2frac{x}{2}} + frac{1-tg^2frac{x}{2}}{1+tg^2frac{x}{2}} right )=\\=frac{2tgfrac{x}{2}-1+tg^2frac{x}{2}}{1+tg^2frac{x}{2}} cdot frac{2tgfrac{x}{2}+1-tg^2frac{x}{2}}{1+tg^2frac{x}{2}} =$

  $=frac{2cdot 0,5-1+0,25}{1+0,25} cdot frac{2cdot 0,5+1-0,25}{1+0,25}$ $=0,2cdot 1,4=0,28$