Answer: 49

Small Hint:

A chord subtending angle $\theta$ has length $2R\sin(\frac{\theta}{2})$

Big Hint:

Let $x=\cos(\frac{\alpha}{2})$ and $y=\cos(\frac{\beta}{2}),$ then eliminate the common radius

Solution:

Put $k=\frac{1}{2R},$ $x=\cos(\frac{\alpha}{2}),$ and $y=\cos(\frac{\beta}{2}).$ The chord data give $$\begin{aligned} \sin\frac\alpha2&=2k,\\ \sin\frac\beta2&=3k,\\ \sin\frac{\alpha+\beta}{2}&=4k. \end{aligned}$$ The addition formula yields $2y+3x=4.$ Also $$\frac{1-x^2}{4}=\frac{1-y^2}{9},$$ so $9x^2-4y^2=5.$ Substituting $y=\frac{4-3x}{2}$ gives $x=\frac{7}{8}.$ Hence $$\cos\alpha=2x^2-1=\frac{17}{32},$$ and the requested sum is $17+32=49.$