Question: finding vector structures in trig expressions

I get trig expressions which are long but contain structure,  The simplest example might be the following which is nothing but a.b where a and b are two unit vectors in polar coordinates.

cos(`θa`)*cos(theta;b)+sin(`θa`)*cos(`φa`)*sin(theta;b)*cos(phi;b)
+sin(`θa`)*sin(`φa`)*sin(theta;b)*sin(phi;b)

Whereas the above is easy to identify, in general the structures are not evident.

Question 1  Is there a way to extract such vectors from long trig expressions? I am most interested in identifying inner, cross and outer products of unit vectors contained in these expressions.

Question 2 I would like to apply trig formulae, double angles etc, selectively.  That is apply the formula to only some of the quantities present, say only on the angles theta;a, and theta;b, but not on the phi's.  How is that done?

Thanks