\( \sin x + \cos x = \sqrt 2 \\ \Leftrightarrow {\left( {\sin x + \cos x} \right)^2} = 2\\ \Leftrightarrow {\sin ^2}x + 2\sin x\cos x + {\cos ^2}x = 1\\ \Leftrightarrow 1 + 2\sin x\cos x = 2\\ \Leftrightarrow 2\sin x\cos x = 1\\ \Leftrightarrow \sin x = \dfrac{1}{{2\cos x}}\\ \text{ Lại có:}{{si}}{{\rm{n}}^2}x + {\cos ^2}x = 1\\ \Rightarrow \dfrac{1}{{4{{\cos }^2}x}} + {\cos ^2}x - 1 = 0\\ \Leftrightarrow 4{\cos ^4}x - 4{\cos ^2}x + 1 = 0\\ \Leftrightarrow {\cos ^2}x = \dfrac{1}{2}\\ \Leftrightarrow \cos x = \pm \dfrac{{\sqrt 2 }}{2}\\ \Rightarrow \sin x = \pm \dfrac{{\sqrt 2 }}{2} \Rightarrow \tan x = 1 \Rightarrow \cot x = 1 \)
