# Simplifying sums of rational trigonometric functions

After simplifying the trigonometric functions as advised before (cf. Simplifying Expressions for FindMinimum), I obtained a sum of 62 similar expressions, the first three as in the below:

(E^(-I (θ[1] + θ[2] + θ[3] + θ[
4] + θ[5]) + I θ[6])
Abs[Sin[ϕ[6]]]^2 (1 -
E^(I (θ[1] - θ[6]))
Cot[ϕ[6]/2] Tan[ϕ[1]/2]) (Cos[θ[1]] +
I Sin[θ[1]] +
E^(I θ[2])
Tan[ϕ[1]/2] Tan[ϕ[2]/2]) (Cos[θ[2]] +
I Sin[θ[2]] +
E^(I θ[3])
Tan[ϕ[2]/2] Tan[ϕ[3]/2]) (Cos[θ[3]] +
I Sin[θ[3]] +
E^(I θ[4])
Tan[ϕ[3]/2] Tan[ϕ[4]/
2]) (Cos[θ[5] - θ[6]] +
I Sin[θ[5] - θ[6]] -
Cot[ϕ[6]/2] Tan[ϕ[5]/2]) (Cos[θ[4]] +
I Sin[θ[4]] +
E^(I θ[5])
Tan[ϕ[4]/2] Tan[ϕ[5]/2]))/(2 Sqrt[(1 +
Cos[ϕ[6]]) (1 + Tan[ϕ[1]/2]^2)]
Sqrt[(1 + Tan[ϕ[1]/2]^2) (1 + Tan[ϕ[2]/2]^2)]
Sqrt[(1 + Tan[ϕ[2]/2]^2) (1 + Tan[ϕ[3]/2]^2)]
Sqrt[(1 + Tan[ϕ[3]/2]^2) (1 + Tan[ϕ[4]/2]^2)]
Sqrt[(1 + Cos[ϕ[6]]) (1 + Tan[ϕ[5]/2]^2)]
Sqrt[(1 + Tan[ϕ[4]/2]^2) (1 + Tan[ϕ[5]/2]^2)])

(E^(-I θ[1])
Abs[Sin[ϕ[1]]] Abs[Sin[ϕ[1]] Sin[ϕ[2]]] Abs[
Sin[ϕ[2]] Sin[ϕ[3]]] Abs[
Sin[ϕ[3]] Sin[ϕ[4]]] Abs[Sin[ϕ[5]]] Abs[
Sin[ϕ[4]] Sin[ϕ[5]]] (Cos[θ[1] - θ[2]] +
Cot[ϕ[1]/2] Cot[ϕ[2]/2] +
I Sin[θ[1] - θ[2]]) (Cos[θ[2] - θ[
3]] + Cot[ϕ[2]/2] Cot[ϕ[3]/2] +
I Sin[θ[2] - θ[3]]) (Cos[θ[3] - θ[
4]] + Cot[ϕ[3]/2] Cot[ϕ[4]/2] +
I Sin[θ[3] - θ[4]]) (Cos[θ[4] - θ[
5]] + Cot[ϕ[4]/2] Cot[ϕ[5]/2] +
I Sin[θ[4] - θ[5]]) (1 -
E^(I (θ[1] - θ[6]))
Cot[ϕ[1]/2] Tan[ϕ[6]/2]) (Cos[θ[5]] +
I Sin[θ[5]] -
E^(I θ[6])
Cot[ϕ[5]/2] Tan[ϕ[6]/2]))/(32 Sqrt[(1 +
Cos[ϕ[1]]) (1 + Cos[ϕ[2]])]
Sqrt[(1 + Cos[ϕ[2]]) (1 + Cos[ϕ[3]])]
Sqrt[(1 + Cos[ϕ[3]]) (1 + Cos[ϕ[4]])]
Sqrt[(1 + Cos[ϕ[4]]) (1 + Cos[ϕ[5]])]
Sqrt[(1 + Cos[ϕ[1]]) (1 + Tan[ϕ[6]/2]^2)]
Sqrt[(1 + Cos[ϕ[5]]) (1 + Tan[ϕ[6]/2]^2)])

(E^(-I (θ[1] + θ[2] + θ[3] + θ[
4] + θ[5]))
Abs[Sin[ϕ[1]]]^2 (Cos[θ[1]] + I Sin[θ[1]] -
E^(I θ[2])
Cot[ϕ[1]/2] Tan[ϕ[2]/2]) (Cos[θ[2]] +
I Sin[θ[2]] +
E^(I θ[3])
Tan[ϕ[2]/2] Tan[ϕ[3]/2]) (Cos[θ[3]] +
I Sin[θ[3]] +
E^(I θ[4])
Tan[ϕ[3]/2] Tan[ϕ[4]/2]) (Cos[θ[4]] +
I Sin[θ[4]] +
E^(I θ[5]) Tan[ϕ[4]/2] Tan[ϕ[5]/2]) (1 -
E^(I (θ[1] - θ[6]))
Cot[ϕ[1]/2] Tan[ϕ[6]/2]) (Cos[θ[5]] +
I Sin[θ[5]] +
E^(I θ[6])
Tan[ϕ[5]/2] Tan[ϕ[6]/2]))/(2 Sqrt[(1 +
Cos[ϕ[1]]) (1 + Tan[ϕ[2]/2]^2)]
Sqrt[(1 + Tan[ϕ[2]/2]^2) (1 + Tan[ϕ[3]/2]^2)]
Sqrt[(1 + Tan[ϕ[3]/2]^2) (1 + Tan[ϕ[4]/2]^2)]
Sqrt[(1 + Tan[ϕ[4]/2]^2) (1 + Tan[ϕ[5]/2]^2)]
Sqrt[(1 + Cos[ϕ[1]]) (1 + Tan[ϕ[6]/2]^2)]
Sqrt[(1 + Tan[ϕ[5]/2]^2) (1 + Tan[ϕ[6]/2]^2)])


At this point, my final goal is to somehow simplify the summation of these expressions. My goal at first was to use the Weierstrass substitution, but it did not amount to something that was easily simplified. In an attempt to simplify the above, I used the function Together to combine denominators, and attempted to apply simplify on the numerator of the function. In particular, analyzing my expressions, I saw that there were patterns, as in the above, where the denominator of the first expression seemed to match with the third expression, so I attempted to combine those two pairwise.

Are there any other things I could try to simplify the summation of these expressions for FindMinimum?

Thanks!

• Out of curiosity, have you already tried to run FindMinimum on these expressions? What is the problem in using the expressions as they are? – MarcoB Aug 13 '15 at 15:30
• That would be a good thing to try first. I thought that I would have to simplify these further because for this sum of 62 functions, I have to add 48 such sets together and there's 96 total variables – user238194 Aug 13 '15 at 16:24
• Well, if you haven't tried it yet, I would strongly urge you to try first. You might be trying to solve a non-existent problem. – MarcoB Aug 13 '15 at 18:27