Question 896411
<pre>
{{{x}}}{{{""=""}}}{{{r(cos(u) + i[""]*sin(u))}}}
{{{y}}}{{{""=""}}}{{{t(cos(v) + i[""]*sin(v))}}}

Multiply equals by equals:

{{{xy}}}{{{""=""}}}{{{r(cos(u) + i[""]*sin(u))*t(cos(v) + i[""]*sin(v))  }}}

{{{xy}}}{{{""=""}}}{{{rt(cos(u) + i[""]*sin(u))*(cos(v) + i[""]*sin(v))  }}}

Use FOIL

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v) + i[""]*cos(u)sin(v) + i[""]*sin(u)cos(v)+i^2*sin(u)sin(v))  }}}

since {{{i^2=-1}}}

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v) + i[""]*cos(u)sin(v) + i[""]*sin(u)cos(v)+(-1)*sin(u)sin(v))  }}}

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v) + i[""]*cos(u)sin(v) + i[""]*sin(u)cos(v)-sin(u)sin(v))  }}}

factor i out of the middle two terms

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v) + i[""](cos(u)sin(v) + sin(u)cos(v))-sin(u)sin(v))  }}}

Move the last term next to the first term:

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v)-sin(u)sin(v)+i[""](cos(u)sin(v) + sin(u)cos(v)))  }}}

Use double angle formulas {{{cos(A+B)=cos(A)cos(B)-sin(A)sin(B)}}}
and {{{sin(A+B)=sin(A)cos(B)+cos(A)sin(B)}}}

You might swap the terms of the imaginary part:

{{{xy}}}{{{""=""}}}{{{rt(cos(u)cos(v)-sin(u)sin(v)+i[""](sin(u)cos(v) + cos(u)sin(v)))  }}}

and you have:

{{{xy}}}{{{""=""}}}{{{rt(cos(u+v)+i[""]sin(u+v))  }}}

Edwin</pre>