= Solution
For a valence-dominated <isoscalar target>, $u_v^N(x)=d_v^N(x)=q_v(x)$. A left-chiral incident <neutrino> has unsuppressed scattering on a left-chiral <quark> and the $(1-y)^2$ weight for a right-chiral <quark>. Summing the up and down contributions yields
$$
\boxed{\frac{d^2\sigma_{\rm NC}^{\nu N}}{dxdy}
=\frac{G_F^2s}{\pi}\,xq_v(x)
\left[G_L^2+G_R^2(1-y)^2\right],}
$$
where $G_L^2=(g_L^u)^2+(g_L^d)^2$ and $G_R^2=(g_R^u)^2+(g_R^d)^2$. The common <normalization> here uses the conventional Fermi coupling and <parton distribution functions>[PDFs] in <6/a>; the requested $y$ shape is independent of an overall <normalization> convention.
Put $w=\sin^2\theta_W$. With $I_3^u=+1/2$, $I_3^d=-1/2$, $Q_u=2/3$ and $Q_d=-1/3$,
$$
g_L^u=\frac12-\frac23w,\quad g_L^d=-\frac12+\frac13w,
\qquad g_R^u=-\frac23w,\quad g_R^d=\frac13w,
$$
so
$$
\boxed{G_L^2=\frac12-w+\frac59w^2,\qquad
G_R^2=\frac59w^2,\qquad
\frac{d\sigma_{\rm NC}}{dy}\propto
\left(\frac12-w+\frac59w^2\right)+\frac59w^2(1-y)^2.}
$$
Fit the relative constant and $(1-y)^2$ components to determine their coupling ratio, or use the <neutrino neutral-to-charged current ratio>[NC-to-CC cross-section ratio] to constrain the absolute combination. In this valence limit, integrating over $0<y<1$ gives
$$
\boxed{R_\nu=\frac{\sigma_{\rm NC}^{\nu N}}{\sigma_{\rm CC}^{\nu N}}
=G_L^2+\frac13G_R^2
=\frac12-w+\frac{20}{27}w^2.}
$$
The same valence <momentum> integral cancels between numerator and denominator. Experimentally, NC events have <hadron>[hadronic] activity without the <weak charged current>[charged-current] outgoing <muon>; known fluxes or CC <normalization>, <muon> identification and acceptance corrections are needed. Sea-quark contributions, charm thresholds, <Quantum chromodynamics>[QCD] corrections and departures from isoscalarity must be included for a precision <weak mixing angle>[weak-mixing-angle] measurement.
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