When composing two rotations, it is useful to know that a rotation about $\alpha$ about an axis $\ell$ can be written as the composition of two reflections in planes containing $\ell$, the first being chosen arbitrarily and the second being at an (oriented) angle $\frac\alpha2$ with respect to the first. Now in the composition of $4$ reflections you get, you can make your choices so that the second and third planes of reflection (the second reflection for the first rotation and the first reflection for the second rotation) are both equal to the unique plane passing through the two axes. Then poof!, those second and third reflections annihilate each other, and you are left with the composition of the first and the fourth reflection, which is a rotation with axis the intersection of those planes, and angle twice the angle between those planes.
If you want to calculate the axis and angle in terms of the original angles, formulas get a bit complicated (even for very easy choices of initial axes as in the question), but such is life, the concrete answer isn't really very easy to write down or remember.
Rodrigues formula is:
$v′=(\cos \theta)v + (\sin \theta)b \times v + (1− \cos \theta) b (b \cdot v)$
Using the fact that:
$b \times (b \times v)=b(b \cdot v)−v(b \cdot b)$
Using $\|b\| = 1$:
$b \times (b \times v)=b(b \cdot v)−v$
$(1− \cos \theta) b \times (b \times v) = (1− \cos \theta)b (b \cdot v) − v + (\cos \theta) v$
Now we need to accomodate those terms in Rodrigues formula:
$v′= v + (\sin \theta)b \times v + (1− \cos \theta ) b (b \cdot v) − v + (\cos \theta) v $
Replacing:
$v′= v + (\sin \theta)b \times v + (1− \cos \theta) b \times (b \times v)$
Now you can apply the matrix identities:
$K v = b \times v$
$K^2 v = b \times (b \times v)$
To get:
$v′= v + (\sin \theta)K v + (1− \cos \theta) K^2 v$
Finally, you can factor out $v$ to get the matrix $R$:
$v′= R v$
$R = I + (\sin \theta)K + (1− \cos \theta) K^2$
Best Answer
$3$D rotation is defined as fixing a pole and rotating the orthogonal sub space to that pole (a unit vector). For instance, if our pole is the vector $(0,0,1)$, we rotate the orthogonal subspace given by the $x-y$ plane.
The sub space is roared according the the rotational matrix.
Defined by:
$$\begin{bmatrix}\cos (\theta) &-\sin (\theta) \\\sin (\theta) & \cos (\theta) \end{bmatrix}$$ .
Choosing basis suitably, we can make $v_1$ our first basis vector and this is fixed by the rotation. While the other bases will be transformed according to our rotation angle. Therefore, all rotation matrices are similar to:
$$\begin{pmatrix} 1&0&0\\ 0&\cos(\theta)&-\sin(\theta)\\ 0&\sin(\theta)&\cos(\theta)\\ \end{pmatrix} $$
Similar matrices have same trace so it follows.
Edit: I should have a book somewhere explaining this in detail, if you want, let me know so that I can find the book and post an image.