ctg-6-pi-9-9-ctg-4-pi-9-11-ctg-2-pi-9-

Question Number 204344 by SEKRET last updated on 13/Feb/24

{ctg}^{6} (\frac{π}{9}) - 9 \cdot {ctg}^{4} (\frac{π}{9}) + 11 \cdot {ctg}^{2} (\frac{π}{9}) = ?

Commented by SEKRET last updated on 14/Feb/24

solution

Commented by Frix last updated on 14/Feb/24

\frac{1}{3}

Answered by Frix last updated on 15/Feb/24

Not as hard as it looks \dots

x = \cot^{2} \frac{π}{9}

c = x^{3} - 9 x^{2} + 11 x

x = t + 3

z = t^{3} - 16 t - 21

t = x - 3 = \cot^{2} \frac{π}{9} - 3 = \frac{2 - 4 \cos \frac{2 π}{9}}{\cos \frac{2 π}{9} - 1} = \frac{2 - 4 c}{c - 1}

z = t^{3} - 16 t - 21 = - \frac{21 c^{3} + c^{2} - 17 c + 3}{c^{3} - 3 c^{2} + 3 c - 1}

c^{3} = \cos^{3} \frac{2 π}{9} = \frac{6 \cos \frac{2 π}{9} - 1}{8} = \frac{6 c - 1}{8}

z = - \frac{\frac{21 (6 c - 1)}{8} + c^{2} - 17 c + 3}{\frac{6 c - 1}{8} - 3 c^{2} + 3 c - 1} =

= - \frac{c^{2} - \frac{5 c}{4} + \frac{3}{8}}{- 3 c^{2} + \frac{15 c}{4} - \frac{9}{8}} = \frac{1}{3}

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