How to solve the trigonometric equation $\sin x + \cos x=\sin 2x + \cos 2x$?

To expand on @gribouillis 's comment, the error in your argument is this step:

$(1-2\cos x)(\sin x+\cos x)=-1$

$\implies (1-2\cos x)=-1$ or $(\sin x +\cos x)=-1$

This is an incorrect implication.

$ab=c$ only implies $a=c$ or $b=c$ when $c=0$.

For $c=-1$ as in this case, you could have $a=1,b=-1$ or $a=5,b=-0.2$ or $a=-1000,b=0.001$ or an infinite number of other combinations.

Use Subtraction: $$\sin 2x−\sin x=\cos x−\cos 2x$$ $$2\sin\frac{x}2\cos\frac{3x}{2}=2\sin\frac{3x}{2}\sin\frac{x}{2}$$ So, $$\sin\frac{x}{2}=0$$ OR $$\tan\frac{3x}{2}=1$$