Determine the general solution of \( \cos x-\sin x=\sqrt{2} \)

Solve the equation \( \cos(x)-\sin(x)=\sqrt{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(x\right)-\sin\left(x\right)=\sqrt{2}\) - step1: Move the expression to the right side: \(-\sin\left(x\right)=\sqrt{2}-\cos\left(x\right)\) - step2: Move the expression to the left side: \(-\sin\left(x\right)-\sqrt{2}=-\cos\left(x\right)\) - step3: Raise both sides to the \(2\)-th power\(:\) \(\left(-\sin\left(x\right)-\sqrt{2}\right)^{2}=\left(-\cos\left(x\right)\right)^{2}\) - step4: Calculate: \(\sin^{2}\left(x\right)+2\sqrt{2}\times \sin\left(x\right)+2=\cos^{2}\left(x\right)\) - step5: Rewrite the expression: \(\sin^{2}\left(x\right)+2\sqrt{2}\times \sin\left(x\right)+2=1-\sin^{2}\left(x\right)\) - step6: Move the expression to the left side: \(\sin^{2}\left(x\right)+2\sqrt{2}\times \sin\left(x\right)+2-\left(1-\sin^{2}\left(x\right)\right)=0\) - step7: Calculate: \(2\sin^{2}\left(x\right)+2\sqrt{2}\times \sin\left(x\right)+1=0\) - step8: Solve using the quadratic formula: \(\sin\left(x\right)=\frac{-2\sqrt{2}\pm \sqrt{\left(2\sqrt{2}\right)^{2}-4\times 2}}{2\times 2}\) - step9: Simplify the expression: \(\sin\left(x\right)=\frac{-2\sqrt{2}\pm \sqrt{\left(2\sqrt{2}\right)^{2}-4\times 2}}{4}\) - step10: Simplify the expression: \(\sin\left(x\right)=\frac{-2\sqrt{2}\pm \sqrt{0}}{4}\) - step11: Simplify the expression: \(\sin\left(x\right)=\frac{-2\sqrt{2}\pm 0}{4}\) - step12: Separate into possible cases: \(\begin{align}&\sin\left(x\right)=\frac{-2\sqrt{2}+0}{4}\\&\sin\left(x\right)=\frac{-2\sqrt{2}-0}{4}\end{align}\) - step13: Simplify the expression: \(\begin{align}&\sin\left(x\right)=-\frac{\sqrt{2}}{2}\\&\sin\left(x\right)=\frac{-2\sqrt{2}-0}{4}\end{align}\) - step14: Simplify the expression: \(\begin{align}&\sin\left(x\right)=-\frac{\sqrt{2}}{2}\\&\sin\left(x\right)=-\frac{\sqrt{2}}{2}\end{align}\) - step15: Evaluate the logic: \(\sin\left(x\right)=-\frac{\sqrt{2}}{2}\) - step16: Use the inverse trigonometric function: \(x=\arcsin\left(-\frac{\sqrt{2}}{2}\right)\) - step17: Calculate: \(\begin{align}&x=\frac{7\pi }{4}\\&x=\frac{5\pi }{4}\end{align}\) - step18: Add the period: \(\begin{align}&x=\frac{7\pi }{4}+2k\pi ,k \in \mathbb{Z}\\&x=\frac{5\pi }{4}+2k\pi ,k \in \mathbb{Z}\end{align}\) - step19: Rewrite the expression: \(x=\left\{ \begin{array}{l}\frac{5\pi }{4}+2k\pi \\\frac{7\pi }{4}+2k\pi \end{array}\right.,k \in \mathbb{Z}\) - step20: Check the solution: \(x=\frac{7\pi }{4}+2k\pi ,k \in \mathbb{Z}\) The general solution of the equation \( \cos x - \sin x = \sqrt{2} \) is \( x = \frac{7\pi}{4} + 2k\pi \), where \( k \) is an integer.