Bài tập nâng cao Phương trình lượng giác cơ bản · Bài 77

$\left\{ \begin{array}{l} \sin x+\sin y=\sqrt{2} \\ \cos x+\cos y=\sqrt{2} \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} \left( \sin x-\cos x \right)+\left( \sin y-\cos y \right)=0 \\ \left( \sin x+\cos x \right)+\left( \sin y+\cos y \right)=2\sqrt{2} \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} \sqrt{2}\sin\left( x-\frac{\pi}{4} \right)+\sqrt{2}\sin\left( y-\frac{\pi}{4} \right)=0 \\ \sqrt{2}\sin\left( x+\frac{\pi}{4} \right)+\sqrt{2}\sin\left( y+\frac{\pi}{4} \right)=2\sqrt{2} \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} \sin\left( x-\frac{\pi}{4} \right)+\sin\left( y-\frac{\pi}{4} \right)=0 \\ \sin\left( x+\frac{\pi}{4} \right)+\sin\left( y+\frac{\pi}{4} \right)=2 \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} \sin\left( x-\frac{\pi}{4} \right)+\sin\left( y-\frac{\pi}{4} \right)=0 \\ \sin\left( x+\frac{\pi}{4} \right)=1 \\ \sin\left( y+\frac{\pi}{4} \right)=1 \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} \sin\left( x-\frac{\pi}{4} \right)+\sin\left( y-\frac{\pi}{4} \right)=0 \\ x=\frac{\pi}{4}+k2\pi \\ y=\frac{\pi}{4}+k2\pi \end{array} \right.$ $\Leftrightarrow \left\{ \begin{array}{l} x=\frac{\pi}{4}+k2\pi \\ y=\frac{\pi}{4}+k2\pi \end{array} \right.$