Question

express cosce(x+iy) as A+iB​

Answer 1

$\cosec(x + yi) = \frac{1}{ \sin(x + yi) }$

$= \frac{1}{ \sin(x) \cos(iy) + \cos(x) \sin(iy) }$

$= \frac{1}{ \sin(x) \cosh(y) + i \cos(x) \sinh(y) }$

$= \frac{ \sin(x) \cosh(y) - i \cos(x) \sinh(y) }{ { \sin }^{2} (x) \cosh ^{2} (x) + { \cos}^{2}(x) { \sinh}^{2}(x) }$

$\sf=( \frac{ \sin(x) \cosh(y) }{ { \sin }^{2} (x) \cosh ^{2} (x) + { \cos}^{2}(x) { \sinh}^{2}(x) } ) - \it{i} \sf ( \frac{\cos(x) \sinh(y) }{ { \sin }^{2} (x) \cosh ^{2} (x) + { \cos}^{2}(x) { \sinh}^{2}(x) } )$

where, sinh(x) and cosh(x) are the hyperbolic sine and cosine functions.