The easiest way to start is to express everything in terms of sine and cosine.
sinA/cosA-cosB/(sinBcosA); sinB/cosB-1/sinA.
We can rewrite these again: (sinAsinB-cosB)/(sinBcosA); (sinBsinA-cosB)/sinAcosB
Now divide: (sinAsinB-cosB)/(sinBcosA) * sinAcosB/(sinBsinA-cosB)=sinAcosB/sinBcosA=tanAcotB QED.