`(cot x - tan x)/(cot 2x)` equals

Options

A. 1
B. 2
C. `-1`
D. 4

Answer

To solve the expression \((\cot x - \tan x) / \cot 2x\), we will follow these steps:

### Step 1: Rewrite \(\tan x\) in terms of \(\cot x\)
We know that:
\[ \tan x = \frac{1}{\cot x} \]
Thus, we can rewrite the expression:
\[ \cot x - \tan x = \cot x - \frac{1}{\cot x} \]

### Step 2: Simplify the numerator
Now, we can combine the terms in the numerator:
\[ \cot x - \frac{1}{\cot x} = \frac{\cot^2 x - 1}{\cot x} \]

### Step 3: Substitute \(\cot 2x\) using the double angle formula
The formula for \(\cot 2x\) is:
\[ \cot 2x = \frac{\cot^2 x - 1}{2 \cot x} \]

### Step 4: Substitute \(\cot 2x\) into the expression
Now we can substitute \(\cot 2x\) into our original expression:
\[ \frac{\frac{\cot^2 x - 1}{\cot x}}{\cot 2x} = \frac{\frac{\cot^2 x - 1}{\cot x}}{\frac{\cot^2 x - 1}{2 \cot x}} \]

### Step 5: Simplify the expression
When we divide by a fraction, we multiply by its reciprocal:
\[ = \frac{\cot^2 x - 1}{\cot x} \cdot \frac{2 \cot x}{\cot^2 x - 1} \]
The \(\cot^2 x - 1\) terms cancel out:
\[ = 2 \]

### Final Answer
Thus, the final answer is:
\[ \frac{\cot x - \tan x}{\cot 2x} = 2 \]

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