Đặt $t=\sin x+\cos x$ ($|t|\le \sqrt2$)
$\Rightarrow \sin^2x+\cos^2x+2\sin x\cos x=t^2$
$\Leftrightarrow \sin x\cos x=\dfrac{t^2-1}{2}$
Phương trình trở thành:
$t+\dfrac{t^2}{2}-\dfrac{1}{2}=1$
$\Leftrightarrow \dfrac{t^2}{2}+t-\dfrac{3}{2}=0$
$\Leftrightarrow t=1$ (TM); $t=-3$ (loại)
$\sin x+\cos x=1$
$\Leftrightarrow \sqrt2\sin(x+\dfrac{\pi}{4})=1$
$\Leftrightarrow \sin(x+\dfrac{\pi}{4})=\dfrac{1}{\sqrt2}$
$\Leftrightarrow x+\dfrac{\pi}{4}=\dfrac{\pi}{4}+k2\pi\Leftrightarrow x=k2\pi$
hoặc $x+\dfrac{\pi}{4}=\dfrac{3\pi}{4}+k2\pi\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi$
