Given that \log _{10} \sin x+\log _{10} \cos x=-1 and that \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), find n.
Given that \log _{10} \sin x+\log _{10} \cos x=-1 and that \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), find n.