试证若z=z＋iy 试证： (1)sin z=sin x．cosh y＋icos x．sinh y；

正确答案：(1)sin z=sin(x+iy) =sinxcos(iy)+cos xsin(iy) =sin xcosh y+icos xsinh y． (2)cos z=cos(x+iy) =cosxcos(iy)一sin xsin(iy) =cos xcosh y+isin xsinh y． (3)｜sin z｜²=｜sin xcosh y+icos xsinh y｜² =sin² xcosh² y+cos² xsinh²y =sin²x(cosh²y—sinh²y)+(cos²z+sin² x)sinh²y =sin²z+sinh²y． (4)｜cos z｜²=｜cos xcosh y—isin xsinh y｜² =cos²xcosh²y一sin²xsinh²y =cos²x(sinh²y+1)+sin²xsinh²y =cos²z+cos²xsinh²y+sin²xsinh²y =cos²x+(cos²x+sin²x)sinh² y =cos²xsinh²y．
(1)sinz=sin(x+iy)=sinxcos(iy)+cosxsin(iy)=sinxcoshy+icosxsinhy．(2)cosz=cos(x+iy)=cosxcos(iy)一sinxsin(iy)=cosxcoshy+isinxsinhy．(3)｜sinz｜2=｜sinxcoshy+icosxsinhy｜2=sin2xcosh2y+cos2xsinh2y=sin2x(cosh2y—sinh2y)+(cos2z+sin2x)sinh2y=sin2z+sinh2y．(4)｜cosz｜2=｜cosxcoshy—isinxsinhy｜2=cos2xcosh2y一sin2xsinh2y=cos2x(sinh2y+1)+sin2xsinh2y=cos2z+cos2xsinh2y+sin2xsinh2y=cos2x+(cos2x+sin2x)sinh2y=cos2xsinh2y．