Cos x+sin x+sin 2x+1=0

Cosx + sinx +sin2x + 1=0
cosx + sinx +2sinxcosx + sin²x+cos²x=0
cosx+sinx + (cos x+ sin x)²=0
(cos x+ sin x)(1+cos x+ sin x)=0

cos x+ sin x=0|: cos x
tgx = -1
x=-π/4 + πk,k ∈ Z

$\cos x+\sin x=-1$

Формула: $a \sin x\pm b\cos x= \sqrt{a^2+b^2} \sin(x\pm\arcsin \frac{b}{\sqrt{a^2+b^2}} )$

$\sqrt{2} \sin(x+ \frac{\pi}{4} )=-1 \\ \sin(x+\frac{\pi}{4})=- \frac{1}{ \sqrt{2} } \\ x+\frac{\pi}{4}=(-1)^k^+^1\cdot \frac{\pi}{4}+\pi k,k \in Z \\ x=(-1)^{k+1}\cdot \frac{\pi}{4}-\frac{\pi}{4}+\pi k,k \in Z$