Limiting Case of Distance Between Points on a Sphere

Question

So, the formula for the angular separation between two points on a sphere is $$\cos d=\sin\delta_1\sin\delta_2+\cos\delta_1\cos\delta_2\cos(\alpha_2-\alpha_1)$$ where $\delta$ denotes the latitude of a point ($\pi-\phi$ in the conventional spherical coordinate system), and $\alpha$ denotes the longitude ($\theta$ in the conventional system). For small distances, because the sphere is a manifold, I thought this should go to the Pythagorean theorem in a small limit. So, neglecting all terms in $O(x^3)$, $$\begin{align}1+\frac{1}{2}d^2 &=\delta_1\delta_2+\left(1+\frac{1}{2}\delta_1^2\right)\left(1+\frac{1}{2}\delta_2^2\right)\left(1+\frac{1}{2}(\alpha_2-\alpha_1)^2\right)\\ &=\delta_1\delta_2+1+\frac{1}{2}\delta_1^2+\frac{1}{2}\delta_2^2+\frac{1}{2}(\alpha_2-\alpha_1)^2\\ d^2&=(\delta_1+\delta_2)^2+(\alpha_2-\alpha_1)^2\end{align}$$ This confuses me very much. Why is it that the deltas add, instead of subtract?

Answer 1

Because $\cos x=1-\frac12x^2+O\left(x^4\right)$, not $+$.