Solution:

\(\displaystyle{\frac{{{\left({\cot{{\left(\theta\right)}}}+{1}\right)}{\left({\cot{{\left(\theta\right)}}}+{1}\right)}}}{{{\csc{\theta}}}}}={\csc{{\left(\theta\right)}}}+{2}{\cos{{\left(\theta\right)}}}\)

Taking LHS:

\(\displaystyle{\frac{{{\left({\cos{\theta}}+{1}\right)}^{{2}}}}{{{\csc{\theta}}}}}={\frac{{{{\cot}^{{2}}\theta}+{2}{\cos{\theta}}+{1}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\frac{{{{\csc}^{{2}}\theta}+{2}{\cos{\theta}}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\frac{{{{\csc}^{{2}}\theta}}}{{{\csc{\theta}}}}}+{\frac{{{2}{\cos{\theta}}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\csc{\theta}}+{\frac{{{2}{\cos{\theta}}}}{{{\sin{\theta}}\cdot{\frac{{{1}}}{{{\sin{\theta}}}}}}}}\)

\(\displaystyle={\csc{\theta}}+{2}{\cos{\theta}}={R}{H}{S}\)

LHS=RHS

Hence proved

\(\displaystyle{\frac{{{\left({\cot{{\left(\theta\right)}}}+{1}\right)}{\left({\cot{{\left(\theta\right)}}}+{1}\right)}}}{{{\csc{\theta}}}}}={\csc{{\left(\theta\right)}}}+{2}{\cos{{\left(\theta\right)}}}\)

Taking LHS:

\(\displaystyle{\frac{{{\left({\cos{\theta}}+{1}\right)}^{{2}}}}{{{\csc{\theta}}}}}={\frac{{{{\cot}^{{2}}\theta}+{2}{\cos{\theta}}+{1}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\frac{{{{\csc}^{{2}}\theta}+{2}{\cos{\theta}}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\frac{{{{\csc}^{{2}}\theta}}}{{{\csc{\theta}}}}}+{\frac{{{2}{\cos{\theta}}}}{{{\csc{\theta}}}}}\)

\(\displaystyle={\csc{\theta}}+{\frac{{{2}{\cos{\theta}}}}{{{\sin{\theta}}\cdot{\frac{{{1}}}{{{\sin{\theta}}}}}}}}\)

\(\displaystyle={\csc{\theta}}+{2}{\cos{\theta}}={R}{H}{S}\)

LHS=RHS

Hence proved