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+(\cos^2x)/(\sin^2x)\right)=2\left((\sin^2x)/(\sin^2x)+(\cos^2x)/(\sin^2x)\right)\\\\=2\left((\sin^2x+\cos^2x)/(\sin^2x)\right)=2\left((1)/(\sin^2x)\right)=(2)/(\sin^2x)\\\\R=2\left((\cos x)/(\sin x)\right)\left((1)/(\cos x)\right)\left((1)/(\sin x)\right)=2\left((1)/(\sin x)\right)\left((1)/(1)\right)\left((1)/(\sin x)\right)\\\\=2\left((1)/(\sin^2x)\right)=(2)/(\sin^2x)`

`\boxed{L=R}`

`Used:\\\\\cot x=(\cos x)/(\sin x)\\\\\sin^2x+\cos^2x=1\\\\\sec x=(1)/(\cos x)\\\\\csc x=(1)/(\sin x)`