Question

докажите тождество sin x + cos x =√2 cos(п/4 - x)
sin x - cos x =-√2 cos(п/4 + x)

Answer 1

$1); ; sinx+cosx=sqrt2cdot ({ frac{1}{sqrt2}cdot sinx +frac{1}{sqrt2}cdot c osx )=$

$=[; sin frac{pi }{4}=cos frac{pi}{4}= frac{sqrt2}{2} =frac{1}{sqrt2}; ] =\\=sqrt2cdot (sinfrac{pi}{4}cdot sinx+cosfrac{pi}{4}cdot cosx)=sqrt2cdot cos(frac{pi}{4}-x)\\\2); ; sinx-cosx=sqrt2cdot ( frac{1}{sqrt2} cdot sinx- frac{1}{sqrt2}cdot cosx)=\\=sqrt2cdot (sin frac{pi}{4} cdot sinx-cosfrac{pi }{4}cdot cosx)=\\=-sqrt2cdot (cosfrac{pi}{4}cdot cosx-sinfrac{pi}{4}cdot sinx)=-sqrt2cdot cos(frac{pi}{4}+x)$