Complex trigonometric functions

Euler´s Formula: e iz = cosz + i sinz; e -iz = cosz - i sinz

Definition:  cosh z = (ez + e-z)/2; sinh z = (ez + e-z)/2  ➙ cosh2z - sinh2z = 1

Complex trigonometric function

e z = e x + iy = ex (cos y + i sin y); e -z = e x - iy = ex (cos y - i sin y);

➙ sin z = 1/2i (e iz - e -iz) = sin x cosh y + i cos x sinh y

cos z = 1/2 (e iz +e -iz) = cos x cosh y - i sin x sinh y

= sin (x + π/2) cosh y + i cos (x + π/2) sinh y

tan z = sin z / cos z = (sin x cosh y + i cos x sinh y) / (cos x cosh y - i sin x sinh y)

Multiplying nominator and denominator with conjugate complex,

and cosh2z - sinh2 z = 1

➙ tan z = (sin x cos x + i sinh y cosh y) / (cos2 x + sinh2 y)