$\tan x+\cot x=2\Rightarrow \frac{\operatorname{sinx}}{\operatorname{cosx}}+\frac {\operatorname{cosx}}{\operatorname{sinx}}=2\Rightarrow \frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\operatorname{sinx}\operatorname{cosx}}=2$  \[{{\sin }^{2}}x+{{\cos }^{2}}x-2\operatorname{sinx}\operatorname{cosx}=0\Rightarrow {{\left( \operatorname{sinx}-\operatorname{cosx} \right)}^{2}}=0\Rightarrow \operatorname{sinx}-\operatorname{cosx}=0\Rightarrow \operatorname{sinx}=\operatorname{cosx}\to x=\frac{\pi }{4}\] $\operatorname{sinx}+\operatorname{cosx}=\sin \left( \frac{\pi }{4} \right)+\cos \left( \frac{\pi }{4} \right)=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=\sqrt{2}$