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 $\textcolor{red}{R H S} .$
  
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  Hope this helps :)