The number of solution of $\sec \mathrm{x} \cos 5\mathrm{x}+1=0$ in the interval $\left[0, 2\mathrm{\pi }\right]$ is

$$\begin{gathered} \frac{\cos 5\mathrm{x}+\cos \mathrm{x}}{\cos \mathrm{x}}=\frac{2 \cos 3\mathrm{x} \cos 2\mathrm{x}}{\cos \mathrm{x}}=0 \\ 2\frac{\left(4{\cos }^3 \mathrm{x}-3 \cos \mathrm{x}\right)}{\cos \mathrm{x}}\left(2 {\cos }^2\mathrm{x}-1\right)=0 \\ 4 {\cos }^2 \mathrm{x}-3=0 \\ \cos \mathrm{x}=\frac{\sqrt{3}}{2} \cos \mathrm{x}=\frac{-\sqrt{3}}{2} \\ \mathrm{x}=\frac{\mathrm{\pi }}{6}, 2\mathrm{\pi }-\frac{\mathrm{\pi }}{6} \mathrm{x}=\frac{5\mathrm{\pi }}{6} \\ \frac{\mathrm{\pi }}{6}, \frac{11\mathrm{\pi }}{6} =\frac{5\mathrm{\pi }}{6}, \frac{7\mathrm{\pi }}{6} \\ 2 {\cos }^2 \mathrm{x}-1=0 \\ \cos \mathrm{x}=\frac{1}{\sqrt{2}} \cos \mathrm{x}=\frac{-1}{\sqrt{2}} \\ \mathrm{x}=\frac{\mathrm{\pi }}{4}, \frac{7\mathrm{\pi }}{4} \\ \mathrm{x}=\frac{3\mathrm{\pi }}{4}, \frac{5\mathrm{\pi }}{4} \end{gathered}$$

Total number of solution $=8$