Consider a Gauss map $N$ on a surface $S$ with Gaussian curvature $K$ everywhere non-zero and with mean curvature $H=0$. The differential of the Gauss map satisfies $$\tag{1} \left(d N_s\right)^2=-K \operatorname{Id}_s. $$ My lecture notes state that it follows from eq. $(1)$ that $\forall s\in S$ and all pairs of tangents $t_1,t_2\in T_s S$ we have that $$\tag{2} dN_s(t_1)\cdot dN_s(t_2)=-K(t_1\cdot t_2) $$ Can someone explain how eq. $(2)$ follows from eq. $(1)$, because I do not see the connection.
Edit: I know that the differential of a Gauss map is self-adjoint. Rewriting eq. $(2)$ with bracket-notation, we then have $$ \left\langle d N_s t_1, d N_s t_2\right\rangle=\left\langle d N_s d N_s t_1, t_2\right\rangle=\left\langle\left(d N_s\right)^2 t_1, t_2\right\rangle=-K\left\langle t_1, t_2\right\rangle $$
Does this seem correct?