I'm trying to fit integral (non-rational) quadratic Bézier curves to circular arcs.
$$ B(t) = (1 - t)^2 P_0 + 2 t (1 - t) P_1 + t^2 P_2 \tag{1} \label{1} $$
Let the angle of the arc be $2\theta$. Assuming the arc starts at
$$P_0 = (r, 0)$$
and ends at
$$P_2 = (r \cos 2\theta, r \sin 2\theta)$$
we intend to find the control point that lies along the angle bisector and thus would be $$ P_1 = (R \cos \theta, R \sin \theta)$$
for some $R$ which is to be solved for. We know the mid point of the arc
$$ B(\frac 1 2) = (r \cos \theta, r \sin \theta) \tag 2 \label 2 $$
Substituting the values of $P_0, P_1, P_2$ and $t = \frac 1 2$ in $\eqref{1}$ and equating values of the $x$ coordinate in $\eqref{1}$ and $\eqref{2}$, we have
$$r \cos \theta = \frac r 4 + \frac {R \cos \theta} 2 + \frac {r \cos 2\theta} 4 $$
Solving for $R$ and simplifying, I could bring it down to
$$ R = \frac {r (4 \cos \theta - \cos 2\theta - 1)} {2 \cos \theta} \tag 3 \label 3 $$
$$ R = r (2 - \frac {\cos 2\theta - 1} {2 \cos \theta}) \tag 4 \label 4 $$
But upon giving the $\eqref 3$ to Wolfrm Alpha it is able to simplify it to
$$ R = r ( 2 - \cos \theta ) $$
but not when $\eqref 4$ is given. I would like to know the identities that were used to simplify it so well; I am unable to arrive it after trying for quite sometime now.
$$\frac{4\cos\theta-\cos2\theta-1}{2\cos\theta}\neq2-\frac{\cos2\theta\color{red}{-}1}{2\cos\theta}$$
You've been sloppy about the minus sign on the right-hand side. In fact,
$$\frac{4\cos\theta-\cos2\theta-1}{2\cos\theta}=2-\frac{\cos2\theta\color{red}{+}1}{2\cos\theta}$$