Here they say that a circle on a unit sphere obtained by fixing $\theta=\frac{\pi}{4}$ and letting $\phi$ vary can be described parametrically as follows $$ \left(\cos\frac{\pi}{4}\cos\pi t, \cos\frac{\pi}{4}\sin\pi t, \sin\frac{\pi}{4}\right) \qquad \text{ where } t\in[-1, 1] $$ How does one obtain this?
Idea 1
I thought maybe they simply computed the radius of the circle $$ r_{\text{circle}} = r_{\text{sphere}} \cdot \sin\theta = \sin \frac{\pi}{4} $$ and then used the parametric form of a circle \begin{align} x &= r_{\text{circle}} \cos \phi \\ y &= r_{\text{circle}} \sin \phi \end{align}
which would give
\begin{align} x &= \sin \frac{\pi}{4}\cos\phi \\ y &= \sin\frac{\pi}{4}\sin\phi \\ z &= r_{\text{sphere}}\cos\theta = \cos\frac{\pi}{4} \end{align}
and setting $\phi = \pi t$ for $t\in[-1, 1]$. However this is different from what they have..
Idea 2
I also tried simply using $\theta = \frac{\pi}{4}$ and $\phi=\pi t$ for $t\in[-1, 1]$ in the parametric equation for a sphere but I get the same result as in "Idea 1".
They do seem to have $\sin$ and $\cos$ mixed up, and your Idea 1 does work. But for what it's worth, $\cos \pi / 4 = \sin \pi / 4$ so technically they're not wrong!