$\begin{array}{l}
Horizontal\,range{\kern 1pt} R = \frac{{{u^2}\sin 2\theta }}{g}\\
For\,angle\,of\,projection\,\left( {{{45}^ \circ } – \theta } \right),\,the\\
horizontal\,range\,is\,\\
\therefore \,{R_1} = \frac{{{u^2}\sin \left[ {2\left( {{{45}^ \circ } – \theta } \right)} \right]}}{g}\\
\,\,\,\,\,\,\,\,\,\,\, = \frac{{{u^2}\,\sin \left( {{{90}^ \circ } – 2\theta } \right)}}{g}\\
\,\,\,\,\,\,\,\,\,\,\, = \frac{{{u^2}\cos \,2\theta }}{g}
\end{array}$

$\begin{array}{l}
For\,angle\,of\,projection\,\left( {{{45}^ \circ } + \theta } \right),\,the\\
horizontal\,range\,is\\
{R_2} = \frac{{{u^2}\sin \left[ {2\left( {{{45}^ \circ } + \theta } \right)} \right]}}{g}\\
\,\,\,\,\,\, = \frac{{{u^2}\,\sin \left( {{{90}^ \circ } + 2\theta } \right)}}{g} = \frac{{{u^2}\cos \,2\theta }}{g}\\
\therefore \,\,\,\,\frac{{{R_1}}}{{{R_2}}} = \frac{{{u^2}\cos 2\theta /g}}{{{u^2}\cos 2\theta /g}} = \frac{1}{1}.\\
\therefore \,The\,range\,is\,the\,same.
\end{array}$