Given,

\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)

Now,

\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)

\(\displaystyle={\sec{{\left({180}+{30}\right)}}}\times{\cot{{\left({360}-{60}\right)}}}+{\sin{{\left({180}+{45}\right)}}}\)

\(\displaystyle=-{\sec{{\left({30}\right)}}}\times{\left(-{\cot{{\left({60}\right)}}}\right)}+{\left(-{\sin{{\left({45}\right)}}}\right)}\)

\(\displaystyle=-{\left({\frac{{{2}}}{{\sqrt{{{3}}}}}}\right)}\times{\left(-{\frac{{{1}}}{{\sqrt{{{3}}}}}}\right)}+{\left({\frac{{{1}}}{{\sqrt{{{2}}}}}}\right)}\)

\(\displaystyle={\frac{{{2}}}{{{3}}}}-{\frac{{{1}}}{{\sqrt{{{2}}}}}}\)

\(\displaystyle={\frac{{{2}\sqrt{{{2}}}+{3}}}{{{3}\sqrt{{{2}}}}}}\)

\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)

Now,

\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)

\(\displaystyle={\sec{{\left({180}+{30}\right)}}}\times{\cot{{\left({360}-{60}\right)}}}+{\sin{{\left({180}+{45}\right)}}}\)

\(\displaystyle=-{\sec{{\left({30}\right)}}}\times{\left(-{\cot{{\left({60}\right)}}}\right)}+{\left(-{\sin{{\left({45}\right)}}}\right)}\)

\(\displaystyle=-{\left({\frac{{{2}}}{{\sqrt{{{3}}}}}}\right)}\times{\left(-{\frac{{{1}}}{{\sqrt{{{3}}}}}}\right)}+{\left({\frac{{{1}}}{{\sqrt{{{2}}}}}}\right)}\)

\(\displaystyle={\frac{{{2}}}{{{3}}}}-{\frac{{{1}}}{{\sqrt{{{2}}}}}}\)

\(\displaystyle={\frac{{{2}\sqrt{{{2}}}+{3}}}{{{3}\sqrt{{{2}}}}}}\)