$\sin 2n+\cos 2n+\sin n=2\cos^2n+\cos n$
$\Leftrightarrow 2\sin n\cos n+2\cos^2n-1+\sin n-2\cos^2n-\cos n=0$
$\Leftrightarrow 2\sin n\cos n-\cos n+\sin n-1=0$
Đặt $t=\sin n-\cos n=\sqrt2\sin (n-\dfrac{\pi}{4})$
$\Leftrightarrow 1-2\sin n\cos n=t^2$
$\Leftrightarrow 2\sin n\cos n=1-t^2$
$PT \Leftrightarrow 1-t^2+t-1=0$
$\Leftrightarrow t^2-t=0$
$\Leftrightarrow t(t-1)=0$
+ Nếu $t=0$:
$\sin(n-\dfrac{\pi}{4})=0$
$\Leftrightarrow n=\dfrac{\pi}{4}+k\pi$
+ Nếu $t=1$:
$\sin(n-\dfrac{\pi}{4})=\dfrac{1}{\sqrt2}$
$\Leftrightarrow n=\dfrac{\pi}{2}+k2\pi$ hoặc $n=\pi+k2\pi$
