$_{\omega =\frac{2\pi }{T}\Rightarrow {{\omega }_{M}}=\frac{2\pi }{0/4\pi }=5\frac{rad}{s},{{\omega }_{N}}=\frac{2\pi }{{{T}_{N}}}=\frac{2\pi }{0/8\pi }=\frac{5}{2}\frac{rad}{s}}^{{{T}_{M}}=0/4\pi s,{{T}_{N}}=2{{T}_{M}}=0/8\pi s}$ 

$F=-kx\Rightarrow ma=-kx\Rightarrow a=-\frac{k}{m}x\xrightarrow{\omega =\sqrt{\frac{k}{m}}}a=-{{\omega }^{2}}x\xrightarrow{{{a}_{M}}={{a}_{N}}}-\omega _{M}^{2}{{x}_{M}}=-\omega _{N}^{2}{{x}_{N}}\xrightarrow[{{x}_{N}}={{A}_{N}}\cos {{\omega }_{N}}t]{{{x}_{M}}={{A}_{M}}\cos {{\omega }_{M}}t}{{A}_{M}}\omega _{M}^{2}\cos {{\omega }_{M}}t={{A}_{N}}\omega _{N}^{2}\cos {{\omega }_{N}}t\xrightarrow{\frac{{{A}_{N}}}{{{A}_{M}}}=\frac{1}{2}}\frac{\cos {{\omega }_{M}}t}{\cos {{\omega }_{N}}t}=\frac{{{A}_{N}}}{{{A}_{M}}}\times \frac{\omega _{N}^{2}}{\omega _{M}^{2}}=\frac{1}{2}\times {{(\frac{\frac{5}{2}}{5})}^{2}}\Rightarrow \frac{\cos (5{{t}_{1}})}{\cos (2/5{{t}_{1}})}=\frac{1}{8}$