Find the Value of (cot x – tan x)/cot 2x
Question:
\[ \frac{\cot x-\tan x}{\cot 2x} \]Solution
First simplify the numerator:
\[ \cot x-\tan x = \frac{\cos x}{\sin x} -\frac{\sin x}{\cos x} \] \[ = \frac{\cos^2 x-\sin^2 x} {\sin x\cos x} \] \[ = \frac{\cos 2x} {\sin x\cos x} \]Using
\[ \sin 2x=2\sin x\cos x \] \[ \cot x-\tan x = \frac{2\cos 2x}{\sin 2x} = 2\cot 2x \]Therefore,
\[ \frac{\cot x-\tan x}{\cot 2x} = \frac{2\cot 2x}{\cot 2x} = 2 \]