Question

Prove that csc x - cot x is the same as (sin x)/(1 + cos x)

Answer 1

`\csc x-\cot x=(\sin x)/(1+\cos x)\\\\L_s=(1)/(\sin x)-(\cos x)/(\sin x)=(1-\cos x)/(\sin x)\\\\R_s=(\sin x)/(1+\cos x)\cdot(1-\cos x)/(1-\cos x)=(\sin x(1-\cos x))/(1^2-\cos^2x) =(\sin x(1-\cos x))/(1-\cos^2x)\\\\=(\sin x(1-\cos x))/(\sin^2x)=(1-\cos x)/(\sin x)\\\\L_s=R_s`


`\text{Used:}\\\\\csc x=(1)/(\sin x)\\\\\cot x=(\cos x)/(\sin x)\\\\a^2-b^2=(a-b)(a+b)\\\\\sin^2x+\cos^2x=1\to\sin^2x=1-\cos^2x`