identita

La scrivo a mano

TAN(x) + TAN(x + 2·pi/3) + TAN(x + 4·pi/3) = 3·TAN(3·x)

1° MEMBRO

analizziamo il 2° ed i 3° termine:

TAN(x + 2·pi/3) = (TAN(x) + TAN(2·pi/3))/(1 - TAN(x)·TAN(2·pi/3))=

=(TAN(x) - √3)/(1 + √3·TAN(x))

TAN(x + 4·pi/3) = (TAN(x) + TAN(4·pi/3))/(1 - TAN(x)·TAN(4·pi/3))=

=(TAN(x) + √3)/(1 - √3·TAN(x))

TAN(x) + (TAN(x) - √3)/(1 + √3·TAN(x)) + (TAN(x) + √3)/(1 - √3·TAN(x))

D(x)= (1 + √3·TAN(x))·(1 - √3·TAN(x)) = 1 - 3·TAN(x)^2

N(x)=

=TAN(x)·(1 - 3·TAN(x)^2) + (TAN(x) - √3)·(1 - √3·TAN(x)) + (TAN(x) + √3)·(1 + √3·TAN(x))=

(TAN(x) = t)

N(t)=t·(1 - 3·t^2) + (t - √3)·(1 - √3·t) + (t + √3)·(1 + √3·t)=

=(t - 3·t^3) + (- √3·t^2 + 4·t - √3) + (√3·t^2 + 4·t + √3)=

=9·t - 3·t^3

N(t)/D(t)=(9·t - 3·t^3)/(1 - 3·t^2)= 3·t·(t^2 - 3)/(3·t^2 - 1)

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2° MEMBRO

3·TAN(3·x) = 3·TAN(x + 2·x)=

=3·(TAN(x) + TAN(2·x))/(1 - TAN(x)·TAN(2·x))

TAN(2·x) = 2·TAN(x)/(1 - TAN(x)^2) = 2·t/(1 - t^2)

quindi:

N(t)/D(t)=3·(t + 2·t/(1 - t^2))/(1 - t·2·t/(1 - t^2))=

3·t·(t^2 - 3)/((t + 1)·(t - 1))/((3·t^2 - 1)/((t + 1)·(t - 1)))=

=3·t·(t^2 - 3)/(3·t^2 - 1)

OK!!!