Reduce \((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\) to the standard form .
\((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\)\(=[\frac{(1+i)-2(1-4i)}{(1-4i)(1+i)}][\frac{3-4i}{5+i}]\)
\(=[\frac{1+i-2+8i}{1+i-4i-4i^2}][\frac{3-4i}{5-3i}]=[\frac{-1+9i}{5-3i}][\frac{3-4i}{5+i}]\)
\([\frac{-3+4i+27i-36^2}{25+5i-15i-3i^2}]=\frac{33+31i-4i}{28-10i}=\frac{33+31i}{14-5i}\)
\(=\frac{(33+3li)}{2(14-5i)}×\frac{(14+5i)}{(14-5i)} \) \([on\,multiplaying\,numerator\,and\,denominator\,by\,(14+5i)]\)
\(\frac{462+165i+434i+155i^2}{2[(14)^2(5i)^2]}=\frac{307+599i}{2(196-25i^2)}\)
\(=\frac{307+599i}{2(221)}=\frac{307+599i}{442}=\frac{307}{442}+\frac{599i}{442}\)
This is the required standard form.