The valur of `(cot x- tan x)/(cot 2x)` is

Options

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

Answer

To find the value of \((\cot x - \tan x) / \cot 2x\), we will simplify the expression step by step.

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot x = \frac{\cos x}{\sin x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Thus, we can rewrite \((\cot x - \tan x)\) as:
\[ \cot x - \tan x = \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \(\sin x \cos x\):
\[ \cot x - \tan x = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x} \]

### Step 3: Use the identity for \(\cos^2 x - \sin^2 x\)
We can use the identity:
\[ \cos^2 x - \sin^2 x = \cos 2x \]
So, we can substitute this into our expression:
\[ \cot x - \tan x = \frac{\cos 2x}{\sin x \cos x} \]

### Step 4: Rewrite \(\cot 2x\)
Now, we need to rewrite \(\cot 2x\):
\[ \cot 2x = \frac{\cos 2x}{\sin 2x} \]

### Step 5: Substitute back into the original expression
Now we substitute \(\cot 2x\) back into the original expression:
\[ \frac{\cot x - \tan x}{\cot 2x} = \frac{\frac{\cos 2x}{\sin x \cos x}}{\frac{\cos 2x}{\sin 2x}} \]

### Step 6: Simplify the expression
This simplifies to:
\[ = \frac{\cos 2x}{\sin x \cos x} \cdot \frac{\sin 2x}{\cos 2x} \]
The \(\cos 2x\) terms cancel out:
\[ = \frac{\sin 2x}{\sin x \cos x} \]

### Step 7: Use the double angle formula for sine
We know that:
\[ \sin 2x = 2 \sin x \cos x \]
So we can substitute this into our expression:
\[ = \frac{2 \sin x \cos x}{\sin x \cos x} \]

### Step 8: Final simplification
The \(\sin x \cos x\) terms cancel out:
\[ = 2 \]

Thus, the final value of \(\frac{\cot x - \tan x}{\cot 2x}\) is:
\[ \boxed{2} \]