2(sinx+cosx)+1+sin2x=0

Решите задачу:

$2(sinx+cosx)+1+sin2x=0$
$2(sinx+cosx)+sin^2x+cos^2x+2sinx*cosx=0$
$2(sinx+cosx)+(sinx+cosx)^2=0$
$(sinx+cosx)(sinx+cosx+2)=0$

$\left[\begin{array}{ccc}sinx+cosx=0\\sinx+cosx=-2\end{array}\right=\ \textgreater \ \left[\begin{array}{ccc}tgx=-1\\sinx+cosx \neq -2\end{array}\right=\ \textgreater \ tgx=-1=\ \textgreater \$
$=\ \textgreater \ x=-\frac{\pi}4+\pi n;n\in Z$