Hint
For the first curve use:
Same considerations for the second one.
Let E, F, G be the coefficients of the first fundamental form of a regular surface S in the parametrization \mathbf{x}:U \subset \mathbb{R}^2 \rightarrow S. Let \varphi (u,v) = c and \psi(u,v)=d be two families of regular curves on \mathbf{x}(U) \subset S. Prove that these two families are orthogonal if and only if
E\varphi_v \psi_v-F(\varphi_u\psi_v+\varphi_v\psi_u) +G\varphi_u\psi_u=0
All I can see is splitting up the terms were we would get (\mathbf{x}_u\varphi_v-\mathbf{x}_v\varphi_u)(\mathbf{x}_u\psi_v-\mathbf{x}_v\psi_u)=0 but that does not go anywhere.
(attached is the tex file and it's pdf build since the latex editor is not rendering)LaTeX1.texLaTeX1.pdf
I don't see the relation. with the consideration of the second curve we would simply have (\varphi_v dv+\varphi_u du)(\psi_v dv+ \psi_u du) = (\varphi_v \psi_v) \langle dv,dv\rangle +\langle dv,du\rangle (\varphi_u\psi_v+\varphi_v\psi_u) +\langle du,du\rangle (\varphi_u\psi_u). But this does not give the desired result