if sin(a+ib)sin(c+id)=1 prove that cosh2btanh2d=cos2a and tanh2bcosh2d=cos2c

Question

hyperbolic function with complex number

Answer 1

e^ix=cos(x)+isin(x); e^-x=e^i(ix)=cos(ix)+isin(ix);

e^-ix=cos(x)-isin(x); e^x=e^-i(ix)=cos(ix)-isin(ix);

cosh(x)=½(e^x+e^-x)=½(cos(ix)-isin(ix)+cos(ix)+isin(ix))=cos(ix);

sinh(x)=½(e^x-e^-x)=½(cos(ix)-isin(ix)-cos(ix)-isin(ix))=-sin(ix).

sin(a+ib)=sin(a)cos(ib)+cos(a)sin(ib)=sin(a)cosh(b)-cos(a)sinh(b);

sin(c+id)=sin(c)cos(id)+cos(c)sin(id)=sin(c)cosh(d)-cos(c)sinh(d).

(sin(a)cosh(b)-cos(a)sinh(b))(sin(c)cosh(d)-cos(c)sinh(d))=1,

sin(a)sin(c)cosh(b)cosh(d)-sin(a)cos(c)cosh(b)sinh(d)-cos(a)sin(c)sinh(b)cosh(d)+cos(a)cos(c)sinh(b)sinh(d)=1.

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