Derivation of multiplication rule.
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Consider the product of quintrights A and B, where:

A = 2 Aq cos 18° + 2 Ar cos 36° + 2 As cos 54° + 2 At cos 72°
B = 2 Bq cos 18° + 2 Br cos 36° + 2 Bs cos 54° + 2 Bt cos 72°

To reduce bulk, write Cn for cos (18n°), giving:

A = 2 AqC1 + 2 ArC2 + 2 AsC3 + 2 AtC4
B = 2 BqC1 + 2 BrC2 + 2 BsC3 + 2 BtC4

Repeated application of the distributive law gives:

 AB = 4 AqBqC1C1 + 4 AqBrC1C2 + 4 AqBsC1C3 + 4 AqBtC1C4 + 4 ArBqC2C1 + 4 ArBrC2C2 + 4 ArBsC2C3 + 4 ArBtC2C4 + 4 AsBqC3C1 + 4 AsBrC3C2 + 4 AsBsC3C3 + 4 AsBtC3C4 + 4 AtBqC4C1 + 4 AtBrC4C2 + 4 AtBsC4C3 + 4 AtBtC4C4

Now apply three trigonometric identities:

 2 cos (x) cos (y) = cos (x + y) + cos (x − y) ⇔ 2 CmCn = Cm + n + Cm − n cos (− x) = cos (+ x) ⇔ C− n = C+ n cos (90° + x) = − cos (90° − x) ⇔ C5 + n = − C5 − n

to obtain:

 AB = 2 AqBq (C2 + C0) + 2 AqBr (C3 + C1) + 2 AqBs (C4 + C2) + 2 AqBt (C5 + C3) + 2 ArBq (C3 + C1) + 2 ArBr (C4 + C0) + 2 ArBs (C5 + C1) + 2 ArBt (− C4 + C2) + 2 AsBq (C4 + C2) + 2 AsBr (C5 + C1) + 2 AsBs (− C4 + C0) + 2 AsBt (− C3 + C1) + 2 AtBq (C5 + C3) + 2 AtBr (− C4 + C2) + 2 AtBs (− C3 + C1) + 2 AtBt (− C2 + C0)

Use of the distributive law to separate and recombine yields:

 AB = 2 C0 (AqBq + ArBr + AsBs + AtBt) + 2 C1 (AqBr + ArBq + ArBs + AsBr + AsBt + AtBs) + 2 C2 (AqBq − AtBt + AqBs + ArBt + AsBq + AtBr) + 2 C3 (AqBr + ArBq − AsBt − AtBs + AqBt + AtBq) + 2 C4 (AqBs + ArBr − ArBt + AsBq − AsBs − AtBr) + 2 C5 (AqBt + ArBs + AsBr + AtBq)

Recall that C5 = 0 and C0 = 1 = 2 (C2C4)

 AB = 4 (AqBq + ArBr + AsBs + AtBt) (C2 − C4) + 2 C1 (AqBr + ArBq + ArBs + AsBr + AsBt + AtBs) + 2 C2 (AqBq − AtBt + AqBs + ArBt + AsBq + AtBr) + 2 C3 (AqBr + ArBq − AsBt − AtBs + AqBt + AtBq) + 2 C4 (AqBs + ArBr − ArBt + AsBq − AsBs − AtBr)

Rearrange:

 AB = 2 C1 (AqBr + ArBq + ArBs + AsBr + AsBt + AtBs) + 2 C2 (AqBq − AtBt + AqBs + ArBt + AsBq + AtBr + 2 AqBq + 2 ArBr + 2 AsBs + 2 AtBt) + 2 C3 (AqBr + ArBq − AsBt − AtBs + AqBt + AtBq) + 2 C4 (AqBs + ArBr − ArBt + AsBq − AsBs − AtBr − 2 AqBq − 2 ArBr − 2 AsBs − 2 AtBt)

Combine similar terms:

 AB = 2 C1 (AqBr + ArBq + ArBs + AsBr + AsBt + AtBs) + 2 C2 (3 AqBq + AtBt + AqBs + ArBt + AsBq + AtBr + 2 ArBr + 2 AsBs) + 2 C3 (AqBr + ArBq − AsBt − AtBs + AqBt + AtBq) + 2 C4 (AqBs − ArBr − ArBt + AsBq − 3 AsBs − AtBr − 2 AqBq − 2 AtBt)