$y=x^2 \cos ^2 2 \pi\left(\frac{\beta \gamma}{\alpha}\right)$

The argument of a trigonometric ratio is always dimensionless.

$\frac{\beta \gamma}{\alpha}=\left[ M ^0 L ^0 T ^0\right] \text { or } \beta \gamma=\alpha \Rightarrow \gamma=\frac{ T }{ L ^2}$

$\text { and } y=x^2 \Rightarrow\left[ L ^2\right]$

$\alpha= s ^{-1} \Rightarrow\left[ T ^{-1}\right], \beta=\left[ LT ^{-1}\right]^{-1} \Rightarrow\left[ L ^{-1} T \right]$

$y=m^2$

$\gamma= ms ^{-2}$