Let $M$ be an n-dimensional hypersurface in $\mathbb R^{n+1}$. For any $x\in \mathbb R^{n+1}$, we define $$ \omega(x)=\frac{x-\langle x, \theta\rangle \theta}{|x-\langle x, \theta\rangle \theta|} $$ where $\langle \cdot, \cdot \rangle$ is Euclidean inner, and $\theta$ is a unit vector. Let $\{ \tau_i \}\subset T_xM$ is orthonormal basis. Then how to show $$ \tau_i[\langle \tau_i,\omega \rangle + \frac{1}{|x-\langle x, \theta\rangle \theta|} \langle x, \tau_i-\langle \tau_i ,\theta\rangle \theta - \langle \tau_i , \omega \rangle \omega \rangle] =\langle H,\omega \rangle + \frac{1}{|x-\langle x, \theta\rangle \theta|} \langle \tau_i , \tau_i \rangle $$ for $x\in M$, where $H$ is mean curvature vector.
Besides, assume $\nu$ is normal vector of $M$, let
$$
k=\langle \nu-\langle \nu ,\theta\rangle \theta - \langle \nu , \omega \rangle \omega , \nu-\langle \nu ,\theta\rangle \theta - \langle \nu , \omega \rangle \omega \rangle
$$
how to show
$$
\langle \tau_i , \tau_i \rangle = n-1-k ~~~
$$
This question is from Proposition 2.6 of 8th page of Mean curvature flow of cylindrical graphs. I don't know how to calculate it , so ask here, thanks.