Angle in Spherical coordinates

[Matterwave]

I have to proove something in QM but I'm stuck on a bit of math.

Say I have two vectors:

$$\vec{a} = (r_a,\theta_a,\phi_a)$$
and
$$\vec{b} = (r_b,\theta_b,\phi_b)$$

What is the cosine of the angle between them? If my proof is to work the cosine of the angle between them have to be:

$$cos(\theta)=1+sin(\theta_a)sin(\theta_b)cos(\phi_a-\phi_b)$$

I think the 1 is erroneous and should be replaced with
$$cos(\theta_a)cos(\theta_b)$$
But I'm not sure and I can't figure out what I did wrong...Which one is it? Is it either?

 

[lippyka]

cos(\theta)=cos(\theta_a)cos(\theta_b)+sin(\theta_a)sin(\theta_b)cos(\phi_a-\phi_b)orcos(\theta)=1+sin(\theta_a)sin(\theta_b)cos(\phi_a-\phi_b)