In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen & Chuang", they claim that Eqn(4.75) is a rotation about the axis along the direction ( ${cos(\pi/8)}$, ${sin (\pi/8)}$, ${cos(\pi/8)}$ ). They then defined an angle ${\theta}$ such that:
${cos(\theta/2)}$ = $cos^2$(${\pi/8}$)
and its also claimed to be an irrational multiple of 2${\pi}$.
We know that the rotational matrices about any arbitrary axis takes the form of
${cos(\theta/2)}$ ${I}$ - ${i}$ (${n_x}$${X}$ + ${n_y}$${Y}$ + ${n_z}$${Z}$) ${sin(\theta/2)}$ ,
but Eqn(4.75) gives:
${cos^2(\pi/8)}$ ${I}$ - ${i}$ [ ${cos(\pi/8)(X+Z)+sin(\pi/8)Y}$ ] ${sin(\pi/8)}$
My question is how does this ${\theta}$ be able to simultaneously satisfy ${sin(\pi/8})$? Why does ${\theta}$ be referenced from ${cos^2(\pi/8)}$ instead of ${sin(\pi/8})$?