Question

Prove that $\:\frac{1 + sinA - cosA}{1 + sinA + cosA}$ $\:= \: \: \sqrt{\frac{ 1 - cosA}{1 + cosA}}$​

Answer 1

$\: \: \: \: \:{\large{\pmb{\sf{\underline{ Here's \: your \: required \: solution!! }}}}}$

Here we are asked to prove :-

• $\sf \green {\:\dfrac{1 + sinA - cosA}{1 + sinA + cosA}}$$\sf \green{\: \: = \: \: \sqrt{\dfrac{ 1 - cosA}{1 + cosA}}}$

⠀⠀⠀ ______________________

$\sf\purple {:\implies \:\dfrac{1 + sinA - cosA}{1 + sinA + cosA}}$

$\sf :\implies \: \: \: \:\dfrac{\dfrac{1}{sinA} + \dfrac{sinA}{sinA} - \dfrac{cosA}{sinA} }{\dfrac{1}{sinA} + \dfrac{sinA}{sinA} + \dfrac{cosA}{sinA} }$

$\sf:\implies \: \: \: \:\dfrac{cosecA + 1 - cotA}{cosecA +1 + cotA}$

$\sf:\implies\: \: \: \:\dfrac{cosecA - cotA + ( {cosec}^{2}A - {cot}^{2}A)}{cosecA + cotA + 1}$

$\sf:\implies\: \: \: \:\dfrac{(cosecA - cotA) + (cosecA + cotA)({cosec}A - {cot}A)}{cosecA + cotA + 1}$

$\sf:\implies \: \: \: \:\dfrac{(cosecA - cotA)(cosecA + cotA + 1)}{cosecA + cotA + 1}$

$\sf:\implies \: \: \: \: \sqrt{ {\bigg(\dfrac{1}{sinA} - \dfrac{cosA}{sinA} \bigg) }^{2} }$

$\sf:\implies \: \: \: \: \sqrt{ {\bigg(\dfrac{1 - cosA}{sinA}\bigg) }^{2} }$

$\sf:\implies \: \: \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{ {sin}^{2}A } }$

$\sf:\implies \: \: \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{1 - {cos}^{2}A } }$

$\sf:\implies\: \: \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{(1 - {cos}A)(1 + cosA) }}$

$\sf \purple {:\implies\: \: \: \: \sqrt{\dfrac{ 1 - cosA}{1 + cosA}}}$

$\qquad\quad\therefore{\underline{\sf{\pmb{\green{Hence,Proved..!!}}}}}$⠀