text: How do you prove (cotA-1)/(cotA+1) = (cosA-sinA)/(cosA+sinA)?
  
  3 Answers
  Apr 7, 2018
  
  Please refer to the Explanation.
  
  Explanation:
  
  $\frac{\cot A - 1}{\cot A + 1}$,
  
  $= \left\{\cos \frac{A}{\sin} A - 1\right\} \div \left\{\cos \frac{A}{\sin} A + 1\right\}$,
  
  $= \left\{\frac{\cos A - \sin A}{\cancel{\sin}} A\right\} \div \left\{\frac{\cos A + \sin A}{\cancel{\sin}} A\right\}$,
  
  $= \frac{\cos A - \sin A}{\cos A + \sin A}$, as desired!
  
  Apr 7, 2018
  
  See below.
  
  Explanation:
  
  $L H S = \frac{\cot A - 1}{\cot A + 1}$
  
  $= \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1}$
  
  $= \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1} \times \sin \frac{A}{\sin} A$
  
  $= \frac{\cos A - \sin A}{\cos A + \sin A}$
  
  $= R H S$
  
  Apr 7, 2018
  
  Refer to explanation.
  
  Explanation:
  
  Starting from $\textcolor{red}{L H S}$
  
  $\frac{\cot A - 1}{\cot A + 1}$
  
  $\Rightarrow \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1}$
  
  Since, $\cot A = \cos \frac{A}{\sin} A$
  
  $\Rightarrow \frac{\frac{\cos A - \sin A}{\sin} A}{\frac{\cos A + \sin A}{\sin} A}$
  
  $\Rightarrow \frac{\frac{\cos A - \sin A}{\cancel{\sin}} A}{\frac{\cos A + \sin A}{\cancel{\sin}} A}$
  
  $\Rightarrow \frac{\cos A - \sin A}{\cos A + \sin A}$
  
  We get $\textcolor{red}{R H S} .$
  
  color(white)(aaaaaaaaaaaaaaaaaaaaĆ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaasasaaaaaaaasaaaaaaaaaaaaaaaaa)
  Hope this helps :)
explanation: The document contains a math question and multiple detailed answers to the question. The question asks how to prove the trigonometric identity \(\frac{\cot A - 1}{\cot A + 1} = \frac{\cos A - \sin A}{\cos A + \sin A}\). Three different answers are provided, each showing a step-by-step proof of the identity.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: How do you prove (cotA-1)/(cotA+1) = (cosA-sinA)/(cosA+sinA)?
  
  3 Answers
  Apr 7, 2018
  
  Please refer to the Explanation.
  
  Explanation:
  
  $\frac{\cot A - 1}{\cot A + 1}$,
  
  $= \left\{\cos \frac{A}{\sin} A - 1\right\} \div \left\{\cos \frac{A}{\sin} A + 1\right\}$,
  
  $= \left\{\frac{\cos A - \sin A}{\cancel{\sin}} A\right\} \div \left\{\frac{\cos A + \sin A}{\cancel{\sin}} A\right\}$,
  
  $= \frac{\cos A - \sin A}{\cos A + \sin A}$, as desired!
  
  Apr 7, 2018
  
  See below.
  
  Explanation:
  
  $L H S = \frac{\cot A - 1}{\cot A + 1}$
  
  $= \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1}$
  
  $= \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1} \times \sin \frac{A}{\sin} A$
  
  $= \frac{\cos A - \sin A}{\cos A + \sin A}$
  
  $= R H S$
  
  Apr 7, 2018
  
  Refer to explanation.
  
  Explanation:
  
  Starting from $\textcolor{red}{L H S}$
  
  $\frac{\cot A - 1}{\cot A + 1}$
  
  $\Rightarrow \frac{\cos \frac{A}{\sin} A - 1}{\cos \frac{A}{\sin} A + 1}$
  
  Since, $\cot A = \cos \frac{A}{\sin} A$
  
  $\Rightarrow \frac{\frac{\cos A - \sin A}{\sin} A}{\frac{\cos A + \sin A}{\sin} A}$
  
  $\Rightarrow \frac{\frac{\cos A - \sin A}{\cancel{\sin}} A}{\frac{\cos A + \sin A}{\cancel{\sin}} A}$
  
  $\Rightarrow \frac{\cos A - \sin A}{\cos A + \sin A}$
  
  We get $\textcolor{red}{R H S} .$
  
  color(white)(aaaaaaaaaaaaaaaaaaaaĆ aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaasasaaaaaaaasaaaaaaaaaaaaaaaaa)
  Hope this helps :)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and multiple detailed answers to the question. The question asks how to prove the trigonometric identity \(\frac{\cot A - 1}{\cot A + 1} = \frac{\cos A - \sin A}{\cos A + \sin A}\). Three different answers are provided, each showing a step-by-step proof of the identity.
  
  Label: multiple-question-and-answer