Question: Do you know why does spherical pythagoream theorem not work here?

Hi! I know that length of blue line and red line equal: Pi/4 and Pi*sin(Pi/8)/4 so from spherical pythagorem theorem green line should equals cos(Pi/4)*cos(Pi*sin(Pi/8)/4) but when I calculate this from formula for length of parametric curve I got something like this (I have just emphasized):

restart;
with(plots);
assume(t, real);
K := plot3d(1, theta = 0 .. 2*Pi, phi = 0 .. Pi, coords = spherical);
S2 := display(spacecurve([cos(t)*sin(t + Pi/8), sin(t + Pi/8)*sin(t), cos(t + Pi/8)], t = 0 .. Pi/4, thickness = 3, color = green)):

int(sqrt(<diff(cos(t)*sin(t + Pi/8), t), diff(sin(t + Pi/8)*sin(t), t), diff(cos(t + Pi/8), t)> . <diff(cos(t)*sin(t + Pi/8), t), diff(sin(t + Pi/8)*sin(t), t), diff(cos(t + Pi/8), t)>), t = 0 .. Pi/4)

-sqrt(2 - sqrt(2))*sqrt(2 + sqrt(2))*sqrt(2)*EllipticE(I/2*sqrt(2 - sqrt(2)), I)*I/2 + sqrt(2 - sqrt(2))*sqrt(2 + sqrt(2))*sqrt(2)*EllipticF(I/2*sqrt(2 - sqrt(2)), I)*I + EllipticE(I/2*sqrt(2 + sqrt(2)), I)*I - 2*I*EllipticF(I/2*sqrt(2 + sqrt(2)), I)

evalf(%)

0.959458136 + 0.*I

evalf(cos(Pi/4)*cos(Pi*sin(Pi/8)/4))

0.6754080210

I am not sure is Maple calculation wrong or I have missed something important?