Question

Упростите выражение
$displaystyle frac{1}{1+cos alpha }-frac{1}{1-sin alpha }\frac{2sin^2 alpha -1}{sin alpha +cos alpha }$

Answer 1

$1); ; frac{1}{1+cosa} - frac{1}{1-sina} = frac{1-sina-1-cosa}{(1+cosa)(1-sina)} = -frac{sina+cosa}{(1+cosa)(1-sina)}\\2); ; frac{2sin^2a-1}{sina+cosa} = frac{-cos2a}{sina+cos} = frac{-(cos^2a-sin^2a)}{sina+cosa} = frac{-(cosa-sina)(cosa+sina)}{sina+cosa} =\\=-(cosa-sina)=sina-cosa=sina-sin(frac{pi}{2}-a)=\\=2sin(a-frac{pi}{4})cdot cosfrac{pi}{4}=sqrt2cdot sin(a-frac{pi}{4})$