Solved: "How to prove the following equation: tan2(π16)+tan2(3π16)+tan2(5π16)+tan2(7π16)=28"

Connor Randall

Answered question

2022-01-29

How to prove the following equation:

\tan^{2} (\frac{π}{16}) + \tan^{2} (\frac{3 π}{16}) + \tan^{2} (\frac{5 π}{16}) + \tan^{2} (\frac{7 π}{16}) = 28

enveradapb

Beginner2022-01-30Added 13 answers

It can also be calculated by applying residue theorem to the function

f (z) = \tan (8 z) \tan^{2} (z)

and the box contour

{R e (z) = 0} \cup {R e (z) = π} \cup {t + \pm \infty i : 0 \leq t \leq π}

. Then,

\int_{R e (z) = 0} f (z) d z

and

\int_{R e (z) = π} f (z) d z

are cancelled.

lim_{y \to \pm \infty} \int_{0}^{π} f (t + i y) d t = \int_{0}^{π} {(\pm i)}^{3} d t = \mp π i

Sum of residues at

z = \frac{2 k + 1}{16} π, k = 0, 1, \dots, 7

is equal to

- \frac{S}{4}

where S is the given sum.
Residue at

z = \frac{π}{2}

is equal to

lim_{z \to \frac{π}{2}} \frac{d}{d z} {(z - \frac{π}{2})}^{2} f (z) = 8

Combining them, by residue theorem we have

2 π i = 2 π i (- \frac{S}{4} + 8)

which gives S=28.

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