(i) `int tan^2 x dx`=____
(ii) `int cot^2 x dx =`______

Answer

Let's solve the integrals step by step.

### Part (i): ∫ tan² x dx

1. **Rewrite tan² x**:
We know that \( \tan^2 x = \sec^2 x - 1 \).
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx \]

2. **Separate the integrals**:
We can separate the integral into two parts:
\[ \int \tan^2 x \, dx = \int \sec^2 x \, dx - \int 1 \, dx \]

3. **Integrate each part**:
- The integral of \( \sec^2 x \) is \( \tan x \).
- The integral of \( 1 \) is \( x \).
\[ \int \tan^2 x \, dx = \tan x - x + C \]

### Final answer for Part (i):
\[ \int \tan^2 x \, dx = \tan x - x + C \]

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### Part (ii): ∫ cot² x dx

1. **Rewrite cot² x**:
We know that \( \cot^2 x = \csc^2 x - 1 \).
\[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx \]

2. **Separate the integrals**:
We can separate the integral into two parts:
\[ \int \cot^2 x \, dx = \int \csc^2 x \, dx - \int 1 \, dx \]

3. **Integrate each part**:
- The integral of \( \csc^2 x \) is \( -\cot x \).
- The integral of \( 1 \) is \( x \).
\[ \int \cot^2 x \, dx = -\cot x - x + C \]

### Final answer for Part (ii):
\[ \int \cot^2 x \, dx = -\cot x - x + C \]

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