Question

use the definition
By definition of the Bernoulli numbers: no
to prove that

Answer 1

We have:

sin z = 2 sin cos

sin x 2 sin - cos

$\csc x=\dfrac 1 2 \csc \dfrac x 2 \sec \dfrac x 2=\dfrac 1 2 \csc \dfrac x 2 \dfrac 2 {e^{i \frac x 2} + e^{-i \frac x 2} }=\displaystyle \csc \dfrac x 2 \dfrac {e^{-i \frac x 2} } {1 + e^{-i x} }$

csc x (1 +e-ir) = csc -e-ig

CSC.X= e-5-csc teir eir - e-ir 16 ix 2ix eir - 1 2 - 1 IF (ix)" Bn (21x)" B (2-n! - n!) 1 Bn (2 (ix)" - (21x)") Bmi"xn-12 (1 - 2n-1) WWW B221 22n-12(1 - 22n-1) (2n)! B2n (-1)"-1,21-12 (22n-1 - 1) (2n)! _

Now, multiplying both sides by and relabelling the index gives us:

8] - B2% (-1)*- * (22k I CSC T = 1 + \ (2)! 2)
$\square$