How do i show this for cos(a+b)?

Have you looked at any of these?

https://goo.gl/HqnAmb

Do you need it to specifically relate to your diagram?

I am having trouble seeing why ...  ?

$\frac{DC}{AD}\cdot\frac{AD}{AB}=sina\cdot cos b$

Make some changes to your diagram.

Now AF = 1, angle AFB = 90 degrees, and the angle at the bottom right corner is called G.

$\quad\cos(a+b)\\ =\dfrac{\text{AC}}{\text{AB}}\\ =\dfrac{\text{AG} - \text{GC}}{\text{AB}}\\ =\dfrac{\text{AG}}{\frac{1}{\cos b}} - \dfrac{\text{GC}}{\frac{1}{\cos b}}\\ =\cos a \cos b - \tan b \sin a \cos b\\ = \cos a \cos b - \sin a \sin b$