Let's say two right angled triangles share a common hypotenuse which measures 10 in length and share an angle which measures $20^\circ$ in total. How do I work out the value of x (the side adjacent to the $20^\circ$ angle)? Using $\cos$ looks like the right strategy to apply but not sure how to proceed...
As has been commented, your question would benefit greatly from a diagram for clarity, but I believe I can answer it anyway.
Recall $\cos(\theta)=\frac{A}{H}$ so rearranging we get $A=H\cos(\theta)$ and now you just have to substitute in the appropriate values. In your diagram, you should be able to see, by symmetry, that the angle inside each triangle will be half of the $20^\circ$ so we have $H=10, \theta=10^\circ$