Question

Se: x - y = 30°, (cosx + cosy)² + (senx + seny)²

Answer 1

$(cosx+cosy)^2+(sinx+siny)^2=$
$cos^2x+2cosxcosy+cos^2y+sin^2x+2sinxsiny+sin^2y$
$2cosxcosx+2sinxsiny+sin^2x+cos^2x+sin^2y+cos^2y$
$2(cosxcosy+sinxsiny)+(sin^2x+cos^2x)+(sin^2y+cos^2y)$

Sabemos que:

$cos(x-y)=cosxcosy+sinxsiny$
e
$sin^2\alpha+cos^2\alpha=1$

Entao:

$2(cosxcosy+sinxsiny)+(sin^2x+cos^2x)+(sin^2y+cos^2y)$
$2(cos(x-y))+1+1$
$2(cos(30))+2$
$2(\frac{\sqrt3}{2})+2$
$\boxed{\sqrt3+2}$