Let $\mathbf{u}$ and $\mathbf{v}$ be two 3D unit vectors. The transform that performs reflection in the plane normal to $\mathbf{u}$ is given by
$$
T_{\mathbf{u}}(\mathbf{x}) = \mathbf{x} – 2(\mathbf{x} \cdot \mathbf{u})\mathbf{u}
$$
and similarly, reflection in the plane normal to $\mathbf{v}$ is performed by
$$
T_{\mathbf{v}}(\mathbf{x}) = \mathbf{x} – 2(\mathbf{x} \cdot \mathbf{v})\mathbf{v}
$$
Let $\theta$ be the angle between $\mathbf{u}$ and $\mathbf{v}$, and let $\mathbf{n}$ be the unit vector in the direction of $\mathbf{u} \times \mathbf{v}$. So, then we know that $\cos\theta = \mathbf{u} \cdot \mathbf{v}$, and $\mathbf{u} \times \mathbf{v} = (\sin\theta)\mathbf{n}$.
The composition of these two reflections is a rotation around $\mathbf{n}$ by an angle of $2\theta$ (I believe), and that rotation is given by Rodrigues' formula:
$$
R(\mathbf{x}) = (\cos2\theta)\mathbf{x} +
(1 – \cos2\theta)(\mathbf{x} \cdot \mathbf{n})\mathbf{n} +
(\sin 2\theta)(\mathbf{x} \times \mathbf{n})
$$
It seems to me that we ought to be able to prove from first principles that
$$
T_{\mathbf{u}}\big( T_{\mathbf{v}}(\mathbf{x}) \big) = R(\mathbf{x})
$$
I've slogged through pages of vector algebra for a few hours, but to no avail. It's depressing — I used to be good at this stuff, but apparently not any more. I'd like a proof that uses nothing but elementary vector arithmetic, and I'd like it to be coordinate-free, please.
Edit
As a couple of people have mentioned, it seems sensible to work in the $\mathbf{u}\text{-}\mathbf{v}\text{-}\mathbf{n}$ coordinate system. This doesn't violate my "coordinate free" requirement as long as we don't start writing out explicit coordinates for $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{n}$. The vector $R(\mathbf{x}) – \mathbf{x}$ should be entirely in the
$\mathbf{u}\text{-}\mathbf{v}$ plane, so all of its $\mathbf{n}$ terms must vanish, and we should be left with an expression that involves only $\mathbf{u}$ and $\mathbf{v}$, which (I hope) will give us the link to the reflections. The algebraic grunt-work involved is what's giving me trouble.
Best Answer
Write $c := \cos\theta$, $s := \sin\theta$, $\mathbf{w} := \mathbf{u}\times\mathbf{v} = s\mathbf{n}$, and $T := T_\mathbf{u}\left(T_\mathbf{v}\right)$, so that we have ...
$$\begin{align} T(\mathbf{x}) &=\mathbf{x}-2(\mathbf{x}\cdot\mathbf{u})\mathbf{u}-2(\mathbf{x}\cdot\mathbf{v})\mathbf{v} + 4c(\mathbf{x}\cdot\mathbf{v})\mathbf{u}\\ R(\mathbf{x}) &= (2c^2-1)\mathbf{x} + 2 s^2 (\mathbf{x}\cdot\mathbf{n}) \mathbf{n} + 2 s c (\mathbf{x}\times\mathbf{n}) \\ &= (2c^2-1)\mathbf{x} + 2 (\mathbf{x}\cdot\mathbf{w})\mathbf{w} + 2 c (\mathbf{x}\times\mathbf{w}) \\ \end{align}$$ Decomposing $\mathbf{x}$ as $p\mathbf{u} + q\mathbf{v} + r \mathbf{w}$, we can get fairly directly ... $$\mathbf{x}\cdot\mathbf{u} = p + q c \qquad \mathbf{x}\cdot\mathbf{v}=pc+q \qquad \mathbf{x}\cdot\mathbf{w}=rs^2 \qquad (\star)$$ $$\mathbf{x}\times\mathbf{w} = \mathbf{x}\times \left(\mathbf{u}\times\mathbf{v}\right) = (\mathbf{x}\cdot\mathbf{v})\mathbf{u}-(\mathbf{x}\cdot\mathbf{u})\mathbf{v} \qquad (\star\star)$$
Then it's straightforward to show that the difference of the transformations vanishes:
$$\begin{align} T(\mathbf{x}) - R(\mathbf{x}) &=\mathbf{x}-2(p+qc)\mathbf{u}-2(pc+q)\mathbf{v} + 4c(pc+q)\mathbf{u}\\ &-\left( (2c^2-1)\mathbf{x} + 2 r s^2 \mathbf{w} + 2 c \left( (pc+q)\mathbf{u} - (p+qc)\mathbf{v} \right) \right) \\[6pt] &= (2-2c^2)\;\mathbf{x} + 2 \left(-p-qc+2pc^2+2qc-pc^2-qc\right)\;\mathbf{u} \\ &+ 2\left(-pc-q+pc+qc^2\right)\mathbf{v} - 2 r s^2 \mathbf{w} \\[6pt] &= 2 s^2 \left( \mathbf{x} - p\mathbf{u} - q \mathbf{v} - r\mathbf{w} \right) \\[6pt] &= 0 \end{align}$$
and we conclude that the transformations are equivalent. $\square$
Edit. Without jumping immediately to the decomposition of $\mathbf{x}$, we can use the expansion in $(\star\star)$ to write
$$\begin{align} \frac{T(\mathbf{x})-R(\mathbf{x})}{2s^2} \;\;&=\;\; \mathbf{x} \;-\; \left( \; \frac{\mathbf{x}.( \mathbf{u} - c \mathbf{v} )}{s^2}\;\mathbf{u} \;+\; \frac{\mathbf{x}.( \mathbf{v} - c \mathbf{u} )}{s^2}\;\mathbf{v} \;+\; \frac{\mathbf{x}.\mathbf{w}}{s^2}\;\mathbf{w} \;\right) \end{align}$$
If you can "see" that the coefficients of $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$ are the components of $\mathbf{x}$ ---which would be clear for orthogonal $\mathbf{u}$ and $\mathbf{v}$, for which $c=0$ and $s=1$--- then you're done. If not, note that you can arrive at this insight by solving the dot-product equations $(\star)$ for $p$, $q$, $r$.