[latex]sin alpha -cos alpha = frac{1}{ sqrt{2}} \ \ (sin alpha -cos alpha)^{2} =( frac{1}{ sqrt{2}} )^{2} \ \ (sin alpha)^{2}-2sin alpha cos alpha+(cos alpha )^{2}= frac{1}{2} \ \ (sin alpha)^{2}+(cos alpha )^{2}-2sin alpha cos alpha= frac{1}{2} \ \ 1-2sin alpha cos alpha= frac{1}{2} \ \ -2sin alpha cos alpha=- frac{1}{2} \ \ sin alpha cos alpha= frac{1}{4} [/latex] [latex]sin alpha+cos alpha =x\ \ (sin alpha +cos alpha)^{2} =x^{2} \ \ (sin alpha)^{2}+2sin alpha cos alpha+(cos alpha )^{2}= x^{2}\ \ (sin alpha)^{2}+(cos alpha )^{2}+2sin alpha cos alpha= x^{2}\ \ 1+2sin alpha cos alpha= x^{2}\ \ 1+frac{1}{4}=x^{2} \ \ x^{2}= frac{5}{4}\\x=frac{sqrt{5}}{2}}\ \sin alpha+cos alpha=frac{sqrt{5}}{2}}[/latex] [latex] left { {sin alpha - cos alpha = frac{1}{ sqrt{2}}} \ \ atop {sin alpha +cos alpha = frac{sqrt{5}}{2}}} ight. \ \ [/latex] [latex]2sin alpha= frac{1}{ sqrt{2}}+ frac{ sqrt{5}}{2}= frac{ sqrt{2}}{2}+frac{ sqrt{5}}{2}= frac{ sqrt{2}+ sqrt{5}}{2} \ \ sin alpha =frac{ sqrt{2}+ sqrt{5}}{4}\ \sinalpha+cosalpha=frac{ sqrt{5}}{2}\ \frac{ sqrt{2}+ sqrt{5}}{4}+cos alpha =frac{ sqrt{5}}{2} \ \ cos alpha =frac{ sqrt{5}}{2} -frac{ sqrt{2}+ sqrt{5}}{4} \ \ cos alpha = frac{2 sqrt{5}- sqrt{2}- sqrt{5}}{4} = frac{ sqrt{5}- sqrt{2}}{4}[/latex] \ \ [/latex]
