$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to Exercise 4.15 from Nielsen & Chuang:
(Composition of single qubit operations) The Bloch representation gives a nice way to visualize the effect of composing two rotations.
(1) Prove that if a rotation through an angle $\beta_1$ about the axis $\hat{n}_1$ is followed by a rotation through an angle $\beta_2$ about an axis $\hat{n}_2$, then the overall rotation is through an angle $\beta_{12}$ about an axis $\hat{n}_{12}$ given by
$$c_{12} = \coefcos \tag{4.19}\label{4.19}$$ $$ s_{12} \hat{n}_{12} = \coefsin, \tag{4.20}\label{4.20} $$ where $c_i = \cos \left( \beta_i/2 \right), s_i = \sin \left( \beta_i/2 \right), c_{12} = \cos \left( \beta_{12}/2 \right),$ and $s_{12}= \sin \left( \beta_{12}/2 \right)$.
Note that $\hat{n}_1$ and $\hat{n}_2$ are real unit vectors in three dimensions (as stated earlier in the book).
I managed to get the expressions on the RHS of \eqref{4.19} and \eqref{4.20} by expanding the rotation matrices into the form $R_{\hat{n}_i} \left( \beta_i \right) = \cos \left( \beta_i/2 \right) I - i \sin \left( \beta_i/2 \right) \hat{n}_i \cdot \vec{\sigma}$ (where $\vec{\sigma}$ denotes the three component vector $(X, Y, Z)$ of Pauli matrices) and taking the product.
Now, assume that $c_{12} \ne 1$. Then I'd expect to recover the unit vector $\hat{n}_{12}$ if I divide \eqref{4.20} through by $s_{12} = \pm \sqrt{1 - c_{12}^2}$:
$$ \hat{n}_{12} = \frac{\coefsin}{ \pm \sqrt{1 - c_{12}^2} } \tag{1} $$
To check that $\hat{n}_{12}$ is of unit length, I take the dot product with itself:
$$ \hat{n}_{12} \cdot \hat{n}_{12} = \frac{s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 + 2 c_1 c_2 s_1 s_2 \cos (\theta) }{1 - c_{12}^2} \tag{2} $$
where $\cos (\theta) = \hat{n}_1 \cdot \hat{n}_2 $. Expanding $c_{12}$ in the denominator, I get:
$$ \hat{n}_{12} \cdot \hat{n}_{12} = \frac{s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 + 2 c_1 c_2 s_1 s_2 \cos (\theta) }{1 - c_1^2 c_2^2 - s_1^2 s_2^2 \cos^2 (\theta) + 2 c_1 c_2 s_1 s_2 \cos (\theta) } \tag{3} $$
which tells me that the following equation should hold in order for the numerator and denominator to balance to 1:
$$ s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 \stackrel{?}{=} 1 - c_1^2 c_2^2 - s_1^2 s_2^2 \cos^2 (\theta) \tag{4}\label{4} $$
However, I don't think \eqref{4} holds in general: $c_i$ and $s_i$ are functions of $\beta_i$. $\theta$ however is a function of $\hat{n}_i$ which I'm free to tune independently of $\beta_i$. So I should be able to conjure up values that violate \eqref{4} easily, which tells me that $\hat{n}_{12}$ is not a unit vector. This seems implausible to me. Where am I making a mistake?
