Question

Prove $\csc x(\sec x-1)-\cot x(1-\cos x)=\tan x-\sin x$

Answer 1

$\cos ecx\left(\sec x-1\right)-\cot x\left(1-\cos x\right)$

$=\frac{1}{\sin x}\left(\frac{1}{\cos x}-1\right)-\frac{\cos x}{\sin x}\left(1-\cos x\right)$

$=\frac{\left(1-\cos x\right)}{\sin x\cos x}-\frac{\cos x\left(1-\cos x\right)}{\sin x}$

$=\frac{\left(1-\cos x\right)-{\cos }^{2}x\left(1-\cos x\right)}{\sin x\cos x}$

$=\frac{\left(1-\cos x\right)\left(1-{\cos }^{2}x\right)}{\sin x\cos x}$

$=\frac{\left(1-\cos x\right)\times {\sin }^{2}x}{\sin x\cos x}$

$=\tan x\left(1-\cos x\right)$

$=\tan x-\tan x\cos x$

$=\tan x-\frac{\sin x}{\cos x}\times \cos x$

$=\tan x-\sin x$