Question

Докажите тождества

а) $frac{cos2a}{1+sin2a} = frac{1-tga}{1+ tga}$

б)$2cos(45 ^{circ} +a)*cos(45 ^{circ} - a) = cos 2a$

Answer 1

$a) frac{cos2 alpha }{1+sin2 alpha }=frac{cos^2 alpha -sin^2 alpha }{sin^2 alpha +cos^2 alpha +2sin alpha cos alpha }=frac{(cos alpha -sin alpha )(cos alpha +sin alpha )}{(cos alpha +sin alpha )^2}=\\=frac{cos alpha -sin alpha }{cos alpha +sin alpha }=frac{cos alpha (1-frac{sin alpha }{cos alpha })}{cos alpha (1+frac{sin alpha }{cos alpha })}=frac{1-tg alpha }{1+tg alpha };$

$b) 2cos(45^circ+ alpha )bulletcos(45^circ- alpha )=\\2bulletfrac{1}{2}(cos((45^circ+ alpha) -(45^circ- alpha ))+cos(45^circ+ alpha +45^circ- alpha ))=\\cos2 alpha +cos90^circ=cos2 alpha +0=cos2 alpha .$