| 1 | α | 1 | α | 1 | α | |||||||
tgα + | tg | +.......+ | tg | = | ctg | −2ctg2α | ||||||
| 2 | 2 | 2n−1 | 2n−1 | 2n−1 | 2n−1 |
| ctg2α − tgαctgα | ctg2α − 1 | |||
tgα = ctgα − ctgα +tgα = ctgα − | = ctgα − | = | ||
| ctgα | ctgα |
| ctg2α − 1 | ||
= ctgα − 2 | = ctgα − 2ctg(2α) | |
| 2ctgα |
| 1 | 1 | |||
L = tgα + .... + | tg (α/2(k−1)) + | tg (α/2k) = // na mocy 2o / | ||
| 2(k−1) | 2k |
| 1 | 1 | |||
= | ctg (α/2(k−1)) − 2ctg(2α) + | tg (α/2k) = | ||
| 2(k−1) | 2k |
| 1 | ||
= | (2ctg (α/2(k−1)) + tg (α/2k)) − 2ctg(2α) = (*) | |
| 2k |
| ctg2 (α/2k) −1 | ctg2x−1 | |||
//ctg(α/2(k−1))= | − wzór na ctg(2x)= | |||
| 2ctg (α/2k) | 2ctgx |
| 1 | 1 | |||
tg (α/2k) = | − wzór tgx = | // | ||
| ctg (α/2k) | ctgx |
| 1 | ctg2 (α/2k) −1 | 1 | ||||
(*) = | ( | + | ) − 2ctg(2α) = | |||
| 2k | ctg (α/2k) | ctg (α/2k) |
| 1 | ctg2 (α/2k) | |||
= | ( | − 2ctg(2α) = | ||
| 2k | ctg (α/2k) |
| 1 | ||
= | ctg (α/2k) − 2ctg(2α) = P | |
| 2k |