1. The solution set of the equation $\frac{\sqrt{\sin ^{2} x+2}+2^{x}}{\sqrt{\sin ^{2} x+2}+2^{x+1}}+\frac{\sqrt{\lg ^{2} x+3}+3^{x}}{\sqrt{\lg ^{2} x+3}+3^{x+1}}+\frac{\sqrt{e^{x}+6}+6^{x}}{\sqrt{e^{x}+6}+6^{x+1}}=1$ is
1. $\phi$

Analysis: When $b>a, m>0$, we have $\frac{a+m}{b+m}>\frac{a}{b}$; hence $\frac{\sqrt{\sin ^{2} x+2}+2^{x}}{\sqrt{\sin ^{2} x+2}+2^{x+1}}>\frac{1}{2}$, $\frac{\sqrt{\lg ^{2} x+3}+3^{x}}{\sqrt{\lg ^{2} x+3}+3^{x+1}}>\frac{1}{3}, \frac{\sqrt{e^{x}+6}+6^{x}}{\sqrt{e^{x}+6}+6^{x+1}}>\frac{1}{6}$, which means the left side of the equation is greater than the right side, so there is no solution.