[tex]\it x\in\Big(0,\ \ \dfrac{\pi}{2}\Big)\ \ \ \ \ \ (*)\\ \\ \\ sinx+cosx=\sqrt2 \Rightarrow (sinx+cosx)^2=(\sqrt2)^2 \Rightarrow sin^2x+2sinxcosx+cos^2x=2\\ \\ \\ \Rightarrow 1+2sinxcosx=2|_{-1} \Rightarrow 2sinxcosx=1\\ \\ \\ (sinx-cosx)^2=sin^2x+cos^2x-2sinxcosx=1-1=0 \Rightarrow sinx-cosx=0 \Rightarrow \\ \\ \\ \Rightarrow sinx=cosx\ \stackrel{(*)}{\Longrightarrow} x=\dfrac{\pi}{4}[/tex]
