Question

Verify 2+2cot^2x=2cotxsecxcscx

Answer 1

$2+2\cot^2x=2\cot x\sec x\csc x\\\\L=2(1+\cot^2x)=2\left(1+\dfrac{\cos^2x}{\sin^2x}\right)=2\left(\dfrac{\sin^2x}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x}\right)\\\\=2\left(\dfrac{\sin^2x+\cos^2x}{\sin^2x}\right)=2\left(\dfrac{1}{\sin^2x}\right)=\dfrac{2}{\sin^2x}\\\\R=2\left(\dfrac{\cos x}{\sin x}\right)\left(\dfrac{1}{\cos x}\right)\left(\dfrac{1}{\sin x}\right)=2\left(\dfrac{1}{\sin x}\right)\left(\dfrac{1}{1}\right)\left(\dfrac{1}{\sin x}\right)\\\\=2\left(\dfrac{1}{\sin^2x}\right)=\dfrac{2}{\sin^2x}$

$\boxed{L=R}$

$Used:\\\\\cot x=\dfrac{\cos x}{\sin x}\\\\\sin^2x+\cos^2x=1\\\\\sec x=\dfrac{1}{\cos x}\\\\\csc x=\dfrac{1}{\sin x}$