Question

Verify the identity tan x + cot x / tan x - cot x = 1/ sin^2x - cos^2x

Answer 1

`(\tan x+\cot x)/(\tan x-\cot x)=(1)/(\sin^2x-\cos^2x)\\\\\text{use}\ \tan x=(\sin x)/(\cos x),\ \cot x=(\cos x)/(\sin x)\\\\\tan x+\cot x=(\sin x)/(\cos x)+(\cos x)/(\sin x)=(\sin x\cdot\sin x)/(\sin x\cos x)+(\cos x\cdot\cos x)/(\sin x\cos x)\\\\=(\sin^2x+\cos^2x)/(\sin x\cos x)\\\\\text{use}\ \sin^2x+\cos^2x=1\\\\=(1)/(\sin x\cos x)\\\\\tan x-\cot x=(\sin x)/(\cos x)-(\cos x)/(\sin x)=(\sin^2x-\cos^2x)/(\sin x\cos x)`

`L_s=(\tan x+\cot x)/(\tan x-\cot x)=((1)/(\sin x\cos x))/((\sin^2x-\cos^2x)/(\sin x\cos x))=(1)/(\sin x\cos x)\cdot(\sin x\cos x)/(\sin^2x-\cos^2x)\\\\=(1)/(\sin^2x-\cos^2x)=R_s\\\\L_s=R_s\Rightarrow The\ identity.`