Question

1+cosA/sinA - sinA/1+cosA = 2cotA
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Answer 1

Question:

$\sf{Prove\:that\:\dfrac{1+cosA}{sinA}-\dfrac{sinA}{1+cosA}=2cotA}$

Identities Used:

$\sf{1-sin^2A=cos^A}\\\\\sf{\dfrac{cosA}{sinA}=cotA}$

Solution:

$\sf{Taking\:L.H.S.}$

$\sf{=\dfrac{1+cosA}{sinA}-\dfrac{sinA}{1+cosA}}$

$\sf{=\dfrac{\left(1+cosA\right)^2-sin^2A}{sinA\left(1+cosA\right)}}$

$\sf{=\dfrac{1+2cosA+cos^2A-sin^2A}{sinA\left(1+cosA\right)}}$

$\sf{=\dfrac{2cos^2A+2cosA}{sinA\left(1+cosA\right)}}$

$\sf{=\dfrac{2cosA\left(1+cosA\right)}{sinA\left(1+cosA\right)}}$

$\sf{=2\dfrac{cosA}{sinA}}$

$\boxed{\sf{2cotA}}$

$\sf{\bold{\textit{HENCE\:PROVED}}}$