\(If a + ib =\frac{(x+1)^2}{2x^2+1}\)
\(=\frac{x^2+i^2+2xi}{2x^2+1}\)
\(=\frac{x^2-1+i2x}{2x^2+1}\)
\(=\frac{x^2-1}{2x^2+1}+1(\frac{2x}{2x^2+1})\)
on comparing real and imaginary parts, we obtain
\(a=\frac{x^2-1}{2x^2+1}\,and\,\,b=\frac{2x}{2x^2+1}\)
\(a^2+b^2=(\frac{x^2-1}{2x^2+1})+(\frac{2x}{2x^2+1})^2\)
\(=\frac{x^4+1-2x^2+4x^2}{(2x+1)^2}\)
\(\frac{x^2+1+2x^2}{(2x^2+1)^2}\)
\(=\frac{(x^2+1)^2}{(2x^2+1)^2}\)
\(=a^2+b^2=\frac{(x^2+1)^2}{(2x+1)^2}\)
Hence, proved.