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 (C2 − C4)
| 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) |