Question Number 96290 by 175 last updated on 31/May/20
$${If}\::\:\mathrm{tan}\left({x}\:+{iy}\right)\:=\:{a}\:+\:{bi}\: \\ $$$${then}\:{find}\:{a},{b} \\ $$
Commented by Tony Lin last updated on 31/May/20
$${tan}\left({x}+{iy}\right) \\ $$$$=\frac{{tanx}+{taniy}}{\mathrm{1}−{tanxtaniy}} \\ $$$$=\frac{{tanx}+{itanhy}}{\mathrm{1}−{tanx}\centerdot{itanhy}}\: \\ $$$$=\frac{\left({tanx}−{tanhyi}\right)\left(\mathrm{1}+{tanxtanhyi}\right)}{\left(\mathrm{1}−{tanxtanhyi}\right)\left(\mathrm{1}+{tanxtanhyi}\right)} \\ $$$$=\frac{{tanx}+{tanxtanh}^{\mathrm{2}} {y}}{\mathrm{1}+{tan}^{\mathrm{2}} {xtanh}^{\mathrm{2}} {y}}+\frac{{tan}^{\mathrm{2}} {xtanhy}−{tanhy}}{\mathrm{1}+{tan}^{\mathrm{2}} {xtanh}^{\mathrm{2}} {y}}{i} \\ $$