Evaluate `intsec ^(2) x * cosec^(2) x dx`

Answer

To evaluate the integral \( \int \sec^2 x \cdot \csc^2 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \sec^2 x \cdot \csc^2 x \, dx \]
Using the definitions of secant and cosecant in terms of sine and cosine, we have:
\[ \sec^2 x = \frac{1}{\cos^2 x} \quad \text{and} \quad \csc^2 x = \frac{1}{\sin^2 x} \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{1}{\cos^2 x \sin^2 x} \, dx \]

### Step 2: Simplify the Expression
We can express the integrand in a more manageable form:
\[ I = \int \frac{1}{\sin^2 x \cos^2 x} \, dx = \int \frac{1}{\sin^2 x \cos^2 x} \cdot \frac{4}{4} \, dx = \int \frac{4}{\sin^2(2x)} \, dx \]
Here, we used the identity \( \sin(2x) = 2 \sin x \cos x \).

### Step 3: Use a Trigonometric Identity
Now, we can rewrite the integral:
\[ I = 4 \int \frac{1}{\sin^2(2x)} \, dx \]
The integral of \( \frac{1}{\sin^2(2x)} \) is a standard integral:
\[ \int \csc^2(2x) \, dx = -\frac{1}{2} \cot(2x) + C \]

### Step 4: Substitute Back
Substituting back into our expression for \( I \):
\[ I = 4 \left(-\frac{1}{2} \cot(2x) + C\right) = -2 \cot(2x) + C \]

### Final Answer
Thus, the evaluated integral is:
\[ \int \sec^2 x \cdot \csc^2 x \, dx = -2 \cot(2x) + C \]