I was trying to prove that:
At a hyperbolic point, the principal directions bissect the asymptotic directions.
Well, I tried to use the Euler's formula:
Being $dN_p$ with eigenvalues $k_1,k_2$, eigeinvectors $e_1,e_2$, take $v$ one asymptotic direction. So, write $v=\cos\theta e_1+\sin \theta e_2$, $\theta $ the angle from $e_1$. We have:
$$II_p(v)=0\iff\\ 0=\cos^2\theta k_1+\sin^2 \theta k_2\iff\\ \sin^2 \theta=\dfrac{-k_1}{k_2-k_1};\cos^2\theta=\dfrac{k_2}{k_2-k_1}.$$
To get bissection, I thought that I should obtain $\sin^2 \theta=\cos^2\theta$. I cannot finish.
Many thanks in advance.
HINT: Solve your equation
$$0 = k_1\cos^2\theta + k_2\sin^2\theta$$ for $\theta$.