Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot y=\cos\theta>0$, where we treat $x$ and $y$ as unit vectors. The angle $\theta$ can be interpreted as the geodesic distance between $x$ and $y$ in the round metric on the sphere $S^{n-1}$.

The Cauchy inequality applied here states that $x\cdot y\le 1$. Equality holds if and only if $x=y$. (The case where $x$ and $y$ are antipodal is ruled out by the condition that the angle $\theta$ is acute.)

I'm curious as how we can sharpen this inequality to account for the situation when the geodesic distance $\theta$ is large. Noting that $\cos \theta < 1-\theta^2/2$ as $\theta$ is acute, this basic estimate geometrically becomes $x\cdot y <1-\theta^2/2$.

I'm wondering if my lower bound is the best possible. In other words, is it true that $1-x\cdot y=O(\theta^2)$ if the angle $\theta$ is acute?

Is there an analogue of this sharpened inequality when we consider points on the unit sphere $S^{2n-1}$ in complex Euclidean space ${\mathbb{C}}^n$ and apply the complex Cauchy inequality instead?