Question

# Establish identity. \frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}

Trigonometric equation and identitie
Establish identity
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\cot{\theta}}}}{{{\csc{\theta}}+{1}}}}$$

2021-09-11
Given identity $$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\cot{\theta}}}}{{{\csc{\theta}}+{1}}}}$$
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\cot{\theta}}}}{{{\csc{\theta}}+{1}}}}$$
First from the left-hand side,
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}$$
Here use the trigonometry reciprocal identity, $$\displaystyle{\csc{\theta}}={\frac{{{1}}}{{{\sin{\theta}}}}}$$ and $$\displaystyle{\cot{\theta}}={\frac{{{1}}}{{{\tan{\theta}}}}}$$
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\frac{{{1}}}{{{\sin{\theta}}}}}-{1}}}{{{\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}}}}$$
$$\displaystyle={\frac{{{\frac{{{1}-{\sin{\theta}}}}{{{\sin{\theta}}}}}}}{{{\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}}}}$$
$$\displaystyle={\frac{{{1}-{\sin{\theta}}}}{{{\cos{\theta}}}}}$$
Multiply $$\displaystyle{\left({1}+{\sin{\theta}}\right)}$$ in numerator and denominator,
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\left({1}-{\sin{\theta}}\right)}{\left({1}+{\sin{\theta}}\right)}}}{{{\cos{\theta}}{\left({1}+{\sin{\theta}}\right)}}}}$$
$$\displaystyle={\frac{{{1}-{{\sin}^{{2}}\theta}}}{{{\cos{\theta}}{\left({1}+{\sin{\theta}}\right)}}}}$$
$$\displaystyle={\frac{{{{\cos}^{{2}}\theta}}}{{{\cos{\theta}}{\left({1}+{\sin{\theta}}\right)}}}}$$
$$\displaystyle={\frac{{{\cos{\theta}}}}{{{\left({1}+{\sin{\theta}}\right)}}}}$$
multiply $$\displaystyle{\frac{{{1}}}{{{\sin{\theta}}}}}$$ in numerator and denominator,
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\cos{\theta}}{\left({\frac{{{1}}}{{{\sin{\theta}}}}}\right)}}}{{{\left({1}+{\sin{\theta}}\right)}{\left({\frac{{{1}}}{{{\sin{\theta}}}}}\right)}}}}$$
$$\displaystyle={\frac{{{1}-{{\sin}^{{2}}\theta}}}{{{\cos{\theta}}{\left({1}+{\sin{\theta}}\right)}}}}$$
$$\displaystyle={\frac{{{{\cos}^{{2}}\theta}}}{{{\cos{\theta}}{\left({1}+{\sin{\theta}}\right)}}}}$$
$$\displaystyle={\frac{{{\cos{\theta}}}}{{{\left({1}+{\sin{\theta}}\right)}}}}$$
multiply $$\displaystyle{\frac{{{1}}}{{{\sin{\theta}}}}}$$ in numerator and denominator,
$$\displaystyle{\frac{{{\csc{\theta}}-{1}}}{{{\cot{\theta}}}}}={\frac{{{\cos{\theta}}{\left({\frac{{{1}}}{{{\sin{\theta}}}}}\right)}}}{{{\left({1}+{\sin{\theta}}\right)}{\left({\frac{{{1}}}{{{\sin{\theta}}}}}\right)}}}}$$
$$\displaystyle={\frac{{{\cot{\theta}}}}{{{\csc{\theta}}+{1}}}}$$
Therefore it is established that the left-hand side equal to the right-hand side quantity.