Question

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

Answer 1

$\csc x-\cot x=\dfrac{\sin x}{1+\cos x}\\\\L_s=\dfrac{1}{\sin x}-\dfrac{\cos x}{\sin x}=\dfrac{1-\cos x}{\sin x}\\\\R_s=\dfrac{\sin x}{1+\cos x}\cdot\dfrac{1-\cos x}{1-\cos x}=\dfrac{\sin x(1-\cos x)}{1^2-\cos^2x} =\dfrac{\sin x(1-\cos x)}{1-\cos^2x}\\\\=\dfrac{\sin x(1-\cos x)}{\sin^2x}=\dfrac{1-\cos x}{\sin x}\\\\L_s=R_s$

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