How can I find $x$ (non-complex number)? Any detailed solution/explanation is welcome :). Thank you in advance.
The equation:
$\sin(x)-\cos(x) = 2\sqrt2\sin(x)\cos(x)$
I'm still new at trigonometry equations, and every new way to solve this will help me a lot.
$$\frac{\sin x-\cos x}{\sqrt2}=\sin\left(x-\frac\pi4\right)$$ and $$2\sin x\cos x=\sin2x,$$ so you are solving $$\sin\left(x-\frac\pi4\right)=\sin2x.$$ But $\sin a=\sin b$ iff either $b-a=2k\pi$ or $a+b=(2k+1)\pi$ for an integer $k$, etc.