Verify the identity:\frac{(\cot(\theta)+1)(\cot(\theta)+1)}{\csc\theta}=\csc(\theta)+2\cos(\theta)

texelaare 2021-09-14 Answered
Verify the identity:
\(\displaystyle{\frac{{{\left({\cot{{\left(\theta\right)}}}+{1}\right)}{\left({\cot{{\left(\theta\right)}}}+{1}\right)}}}{{{\csc{\theta}}}}}={\csc{{\left(\theta\right)}}}+{2}{\cos{{\left(\theta\right)}}}\)

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Expert Answer

comentezq
Answered 2021-09-15 Author has 10515 answers
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
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