Prove that Equation without resolution in R :
I am not really sure what you are asking but I don't think that equation holds true anywhere.
$\displaystyle
\cos (u) = \sin (v)
$
only when $\displaystyle
u = v + \pi /2
$
$\displaystyle
\begin{gathered}
u = \sin (x) \hfill \\
v = \cos (y) \hfill \\
0 \leqslant u \leqslant 1 \hfill \\
0 \leqslant v \leqslant 1 \hfill \\
u \ne v + \pi /2 \hfill \\
\end{gathered}
$