`int(tanx-cotx)^(2)dx=`

Options

A. `tanx+cotx+c`
B. `tanx-cotx+c`
C. `-tanx-cotx+c`
D. `-tanx+cotx+c`

Answer

To solve the integral \( \int (\tan x - \cot x)^2 \, dx \), we can follow these steps:

### Step 1: Expand the integrand
First, we expand the expression \( (\tan x - \cot x)^2 \):
\[ (\tan x - \cot x)^2 = \tan^2 x + \cot^2 x - 2 \tan x \cot x \]

### Step 2: Substitute cotangent
Recall that \( \cot x = \frac{1}{\tan x} \). Therefore, we can rewrite the expression:
\[ \tan^2 x + \cot^2 x - 2 \tan x \cot x = \tan^2 x + \frac{1}{\tan^2 x} - 2 \]

### Step 3: Use trigonometric identities
We can use the identity \( \tan^2 x = \sec^2 x - 1 \) and \( \cot^2 x = \csc^2 x - 1 \):
\[ \tan^2 x + \cot^2 x = (\sec^2 x - 1) + (\csc^2 x - 1) = \sec^2 x + \csc^2 x - 2 \]
Thus, we have:
\[ \tan^2 x + \cot^2 x - 2 = \sec^2 x + \csc^2 x - 4 \]

### Step 4: Rewrite the integral
Now we can rewrite the integral:
\[ \int (\tan x - \cot x)^2 \, dx = \int (\sec^2 x + \csc^2 x - 4) \, dx \]

### Step 5: Separate the integrals
We can separate the integral into three parts:
\[ \int \sec^2 x \, dx + \int \csc^2 x \, dx - 4 \int dx \]

### Step 6: Integrate each part
Now we can integrate each part:
1. \( \int \sec^2 x \, dx = \tan x + C_1 \)
2. \( \int \csc^2 x \, dx = -\cot x + C_2 \)
3. \( \int dx = x + C_3 \)

Putting it all together:
\[ \int (\tan x - \cot x)^2 \, dx = \tan x - \cot x - 4x + C \]

### Final Answer
Thus, the final result is:
\[ \int (\tan x - \cot x)^2 \, dx = \tan x - \cot x - 4x + C \]