If you just want to prove that exterior angle COB = 2 x angle A,

consider a line AO and extend it to D.

AOD divides angle A into A1 (CAO) and A2 (BAO).

by theorem of exterior angles of triangles, you see that

angles COD = 2x A1 and angles BOD = 2 x A2

hence exterior angle COB = 2 x (A1 + A2) = 2A