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 :)