Problem 1.\ (a) Find the reciprocal of

x+iy

, working in polar form but expressing the\ final result in Cartesian form.\ (b) Using the identities\

cosz=(e^(iz)+e^(-iz))/(2),sinz=(e^(iz)-e^(-iz))/(2)

\ established from comparison of power series, show that\ i.\

sin(x+iy)=sinxcoshy+icosxsinhy,\ cos(x+iy)=cosxcoshy-isinxsinhy,

\ ii.

|sinz|^(2)=sin^(2)x+sinh^(2)y,|cosz|^(2)=cos2x+sinh^(2)y

.

Problem 1. (a) Find the reciprocal of x+iy, working in polar form but expressing the final result in Cartesian form. (b) Using the identities cosz=2eiz+eiz,sinz=2eizeiz established from comparison of power series, show that i. sin(x+iy)=sinxcoshy+icosxsinhy,cos(x+iy)=cosxcoshyisinxsinhy, ii. sinz2=sin2x+sinh2y,cosz2=cos2x+sinh2y