Let $v, w \in \mathbb{R}^d$ and denote $\theta = \angle(v, w) = \arccos \frac{v^\top w}{\lVert v \rVert_2 \lVert w \rVert_2} \in [0, \pi]$. Let $\xi \sim N(0, \sigma^2 I)$ be a Gaussian random vector in $\mathbb{R}^d$. Denote $\theta_\xi = \angle(v, w+\xi)$.
I empirically find that $$\mathbb{E}_\xi \left[(\pi-\theta_\xi)\cos\theta_\xi+\sin\theta_\xi\right]$$ is monotone w.r.t $\theta$, and for a sufficiently large $\sigma^2$, it is larger than 1 (approximately $1+O(\frac{1}{d})$) for all $\theta$.
I am stuck at finding theoretical justifications of such results. Can anyone help?
Notice that
\begin{align*} \theta_\xi \stackrel{d}{=} \angle (v, \sigma^{-1}w + \xi_1), \end{align*}
where $\xi_1 \stackrel{d}{=} \sigma^{-1}\xi \sim \mathcal{N}(0,I)$. As $\sigma\to\infty$, this converges in distribution to $\angle(v, \xi_1)$. Since $\xi_1/\|\xi_1\|$ is uniformly distributed over the unit sphere $S^{d-1}$, the limiting distribution has density
\begin{align*} f(\theta) := \frac{\Gamma(\frac{d}{2})}{\sqrt{\pi}\Gamma(\frac{d-1}{2})}\sin^{d-2}\theta \mathbf{1}_{[0,\pi]}{\theta} \end{align*}
and hence
\begin{align*} \lim_{\sigma\to\infty} \mathbb{E}\left[(\pi-\theta_\xi)\cos\theta_\xi+\sin\theta_\xi\right] &= \int_{0}^{\pi} \left[ (\pi - \theta)\cos\theta + \sin\theta \right] f(\theta) \, d\theta \\ &= \frac{\Gamma(\frac{d}{2})\Gamma(\frac{d+2}{2})}{\Gamma(\frac{d+1}{2})} \\ &= 1 + \frac{1}{2d} + \frac{1}{8d^2} + \cdots \quad \text{as } d \to \infty. \end{align*}