The number of solution(s) of the equation $\sec \mathrm{x}+\mathrm{cosec} \mathrm{x}=2\sqrt{2}\text{ in }[0,8\mathrm{\pi }]$ is/are

$$\begin{gathered} \begin{array}{l}\frac{1}{\cos \mathrm{x}}+\frac{1}{\sin \mathrm{x}}=2\sqrt{2}\\ \sin \mathrm{x}+\cos \mathrm{x}=\sqrt{2}\sin 2\mathrm{x}\\ \frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\sin 2\mathrm{x}\\ \sin 2x=\sin \left(x+\frac{\mathrm{\pi }}{4}\right)\Rightarrow 2x=n\mathrm{\pi }+{\left(-1\right)}^n\left(x+\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow x\left(2-{\left(-1\right)}^n\right)=\left(n+\frac{{\left(-1\right)}^n}{4}\right)\mathrm{\pi } \\ \Rightarrow \mathrm{x}=\frac{\left(\mathrm{n}+\frac{{\left(-1\right)}^{\mathrm{n}}}{4}\right)\mathrm{\pi }}{\left(2-{\left(-1\right)}^{\mathrm{n}}\right)}\\ number of solutions in \left[0,2\mathrm{\pi }\right] is 3 \left(by taking n=1,3,5\right) \\ \therefore number of solutions in \left[0,8\mathrm{\pi }\right] is 12, \\ \left( \mathrm{since} secx+\cos ecx \mathrm{is} \mathrm{a} \mathrm{periodic} \mathrm{with} \mathrm{period} 2\mathrm{\pi }\right)\\ \end{array} \end{gathered}$$