If \[\dfrac{{{{\left( {1 + i} \right)}^2}}}{{2 - i}} = x + iy\], then find the value of \[x + y\].
A. \[\dfrac{{ - 2}}{5}\]
B. \[\dfrac{6}{5}\]
C. \[\dfrac{2}{5}\]
D. \[\dfrac{{ - 6}}{5}\]

Hint: In this question, for determining the value of \[x + y\], we need to simplify the expression \[\dfrac{{{{\left( {1 + i} \right)}^2}}}{{2 - i}} = x + iy\] and after that we will compare real and imaginary parts of a complex number to determine the value of \[x + y\].


Complete step by step answer: We know that \[\dfrac{{{{\left( {1 + i} \right)}^2}}}{{2 - i}} = x + iy\]
Let us simplify the above expression.
We know that \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
Thus, we get
\[\dfrac{{1 + 2i + {i^2}}}{{2 - i}} = x + iy\]
Put \[{i^2} = - 1\] in the above expression.
\[\begin{gathered}
  \dfrac{{1 + 2i + \left( { - 1} \right)}}{{2 - i}} = x + iy \\
   \Rightarrow \dfrac{{2i}}{{2 - i}} = x + iy \\
\end{gathered} \]
Now, multiply by \[\left( {2 + i} \right)\] to numerator and denominator.
\[\begin{gathered}
   \Rightarrow \dfrac{{2i\left( {2 + i} \right)}}{{\left( {2 - i} \right)\left( {2 + i} \right)}} = x + iy \\
   \Rightarrow \dfrac{{2i\left( {2 + i} \right)}}{{\left( {{2^2} - {i^2}} \right)}} = x + iy \\
   \Rightarrow \dfrac{{4i + 2{i^2}}}{{\left( {4 - \left( { - 1} \right)} \right)}} = x + iy \\
   \Rightarrow \dfrac{{4i + 2\left( { - 1} \right)}}{{\left( {4 - \left( { - 1} \right)} \right)}} = x + iy \\
\end{gathered} \]
By simplifying, we get
\[\begin{gathered}
   \Rightarrow \dfrac{{4i - 2}}{{\left( 5 \right)}} = x + iy \\
   \Rightarrow \dfrac{{ - 2 + 4i}}{{\left( 5 \right)}} = x + iy \\
   \Rightarrow \dfrac{{ - 2}}{{\left( 5 \right)}} + \dfrac{{4i}}{5} = x + iy \\
\end{gathered} \]
By comparing real part and imaginary part with the right-handed sided complex number, we get
\[x = - \dfrac{2}{5}\]and \[y = \dfrac{4}{5}\]
Let us find the value of \[x + y\]
\[\begin{gathered}
  x + y = \dfrac{{ - 2}}{5} + \dfrac{4}{5} \\
   \Rightarrow x + y = \dfrac{{ - 2 + 4}}{5} \\
   \Rightarrow x + y = \dfrac{2}{5} \\
\end{gathered} \]
Hence, the value of \[x + y\] is \[\dfrac{2}{5}\].

Therefore, the option (C) is correct

Note: Many students make mistakes in solving the calculation part and applying trigonometric identities. This is the only way, through which we can solve the example in the simplest way. Using proper trigonometric identities is necessary for solving trigonometric problems as this makes them simple.