`cotx-tanx=?`

Options

A. `cot2x`
B. `2cot^(2)x`
C. `2cot2x`
D. `cot^(2)2x`

Answer

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

### Step 1: Write the definitions of cotangent and tangent
The cotangent and tangent functions can be expressed in terms of sine and cosine:
\[ \cot x = \frac{\cos x}{\sin x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]

### Step 2: Substitute the definitions into the expression
Substituting these definitions into the expression gives:
\[ \cot x - \tan x = \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \]

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of \( \sin x \) and \( \cos x \) is \( \sin x \cos x \). Thus, we can rewrite the expression as:
\[ \cot x - \tan x = \frac{\cos^2 x}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x} \]

### Step 4: Combine the fractions
Now that we have a common denominator, we can combine the fractions:
\[ \cot x - \tan x = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x} \]

### Step 5: Use the double angle identity
We recognize that \( \cos^2 x - \sin^2 x \) can be expressed using the double angle identity:
\[ \cos^2 x - \sin^2 x = \cos 2x \]
Thus, we can rewrite the expression as:
\[ \cot x - \tan x = \frac{\cos 2x}{\sin x \cos x} \]

### Step 6: Use another double angle identity
The denominator \( \sin x \cos x \) can also be expressed using the double angle identity:
\[ \sin x \cos x = \frac{1}{2} \sin 2x \]
So we can rewrite the expression as:
\[ \cot x - \tan x = \frac{\cos 2x}{\frac{1}{2} \sin 2x} = \frac{2 \cos 2x}{\sin 2x} \]

### Step 7: Simplify to cotangent
Finally, we can express this in terms of cotangent:
\[ \cot x - \tan x = 2 \cot 2x \]

Thus, the final result is:
\[ \cot x - \tan x = 2 \cot 2x \]

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