Let us consider (in spherical coordinates) the expression 

[Great arc distance between two points on a unit sphere](https://math.stackexchange.com/questions/231221/great-arc-distance-between-two-points-on-a-unit-sphere)

$$d({\bf v}\_1,{\bf v}\_2)=\cos^{-1}\left(\cos\theta\_1\cos\theta\_2+\sin\theta\_1\sin\theta\_2\cos\left(\varphi\_1-\varphi\_2\right)\right) \qquad (1)$$
that gives the geodesic distance between two points
$${\bf v}\_i=(\sin\theta\_i\cos\varphi\_i,\sin\theta\_i\sin\varphi\_i,\cos\theta\_i)$$
on the unit sphere $S$.

A question:

Assume $\,{\bf v}\_i\in C\,$ where $\,C\,$ is the circle $\,
\left\{\theta=\pi/4 \right\}\cap S\,$. The pole (the origin of the spherical coordinate system) is still the center of the sphere of course.

Then $\,\theta\_1=\theta\_2=\frac{\pi}{4}\,$ and, by (1), we have 
$$d({\bf v}\_1,{\bf v}\_2)=\cos^{-1}\left(\frac{1}{2}+\frac{1}{2}\cos\left(\varphi\_1-\varphi\_2\right)\right)=\cos^{-1}\left(\frac{1}{2}+\frac{1}{2}\cos\left(\varphi\_1-\varphi\_2\right)\right)=\\ =\cos^{-1}\left(\cos^{2}\frac{\varphi\_1-\varphi\_2}{2}\right)$$.

On the other hand, the radius of $C$ is $\frac{1}{2}$. So the arc length between $\,{\bf v}\_1,\,{\bf v}\_2\,$ should also be 
$$\text{radius of}\;\;C\times |\varphi\_1-\varphi\_2|=
\frac{|\varphi\_1-\varphi\_2|}{2}$$

My problem is 
$$\cos^{-1}\left(\cos^{2}\frac{\varphi\_1-\varphi\_2}{2}\right)\not\equiv \frac{|\varphi\_1-\varphi\_2|}{2}$$
Take for instance $\varphi\_1=\pi/2$, $\varphi\_2=0$. Then 
$\cos^{-1}\left(\cos^{2}\frac{\varphi\_1-\varphi\_2}{2}\right)=\frac{\pi}{3}$
while 
$\frac{\varphi\_1-\varphi\_2}{2}=\frac{\pi}{4}$.

Where am I mistaken ?

Thanks