Question

Доказать тождества:
$frac{sinx+sin3x}{cosx+cos3x}=tg2x$
$frac{sin2 alpha +sin5 alpha -sin3 alpha }{cos alpha +1-2sin^22 alpha } =2sin alpha$

Answer 1

$frac{sinx+sin3x}{cosx+cos3x}$ формула суммы 
$sinx+sin3x=2sin2x*cosx\ cosx+cos3x=2cosx*cos2x\ \ frac{ 2sin2x*cosx}{2cos2x*cosx}=frac{sin2x}{cos2x}=tg2x$ верно  
$frac{ sin2a+sin5a-sin3a}{cosa+1-2sin^22a}=\ frac{sin2a+2sina*cos4a}{cosa+1-8sin^2a*cos^2a}=\ frac{2sina*cosa+2sina*cos4a}{cosa+1-8sin^2a*cos^2a}=\ \ frac{2sina(cosa+cos4a)}{cosa+1-8sin^2a*cos^2a}=\\ cos4a=cos^22a-sin^22a=2cos^22a-1=2cos2a*cos2a-1\ 2(2cos^2a-1)^2-1=2(4cos^4a-4cos^2a+1)-1=8cos^4a-8cos^2a+1\ 8cos^2a(cos^2a-1)+1=-8cos^2a*sin^2a+1\ \ frac{2sina*(cosa-8cos^2a*sin^2a+1)}{cosa+1-8sin^2a*cos^2a}=2sina$ 
 верно