The equation $\mathrm{y}={\mathrm{acos}}^2(2\mathrm{\pi nt}-2\mathrm{\pi x}/\mathrm{\lambda })$ represents a wave with

Given wave is 

$$\begin{gathered} \mathrm{y}={\mathrm{acos}}^2(2\mathrm{\pi nt}-2\mathrm{\pi x}/\mathrm{\lambda }) \\ 2{\cos }^2\mathrm{\theta }=\cos 2\mathrm{\theta }+1 \\ \mathrm{or} {\cos }^2\mathrm{\theta }=(\cos 2\mathrm{\theta }+1)/2 \\ \mathrm{y}=\mathrm{a}\left[\frac{\cos 2(2\mathrm{\pi nt}-2\mathrm{\pi x}/\mathrm{\lambda })+1}{2}\right] \\ =\frac{\mathrm{a}}{2}\left[\cos \left(4\mathrm{\pi nt}-\frac{4\mathrm{\pi x}}{\mathrm{\lambda }}\right)+1\right] \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{a} \mathrm{wave} \mathrm{is} \mathrm{given} \mathrm{by} \\ \mathrm{y}={\mathrm{a}}^{\mathrm{\prime }}\left[\cos \left({\mathrm{\omega }}^{\mathrm{\prime }}\mathrm{t}-\frac{2\mathrm{\pi x}}{{\mathrm{\lambda }}^{\mathrm{\prime }}}\right)+1\right] \\ {\mathrm{a}}^{\mathrm{\prime }}=\frac{\mathrm{a}}{2},\mathrm{\omega t}=4\mathrm{\pi nt} \\ \mathrm{or} 2{\mathrm{\pi n}}^{\mathrm{\prime }}\mathrm{t}=4\mathrm{\pi nt}\text{ or }{\mathrm{n}}^{\mathrm{\prime }}=2\mathrm{n} \\ \mathrm{Further} \frac{2\mathrm{\pi x}}{{\mathrm{\lambda }}^{\mathrm{\prime }}}=\frac{4\mathrm{\pi x}}{\mathrm{\lambda }}\text{ or }{\mathrm{\lambda }}^{\mathrm{\prime }}=\mathrm{\lambda }/2 \end{gathered}$$