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

In an expression I have several terms with different coefficients and permutations of them say

aS⁺bc + bS⁺cS^- + cb + others etc

I want to order them the way I want, like, say

abcS⁺ + bcS⁺cS^- +bc  etc

How can I do this?

Happy New Year!

In my expresions I have an integer, nx, which actually has values of only +1 and -1 but I do not specify which.

THe results come out as powers of nx, say nx^n, where n is a positive integer.

How do I reduce the expression nx^n,= 1 for n even and nx^n,= nx for n odd?

I am interested in obtaining a Maple program (symbolic language) that can evaluate exponentials of non-commuting operators. 

e^A+B=e^Ae^Be^C₁e^C₂e^C₃...

See R. M. Wilcox, J. Math. Phys. 8, (1967), 962

This is the Baker-Campbell-Hausdorff expansion, or equivalently the Zassenhaus expansion.