To solve the integral [tex]\(\int \sec^2 x \cdot \csc^2 x \, dx\)[/tex], let's break down the integral step-by-step:
1. Identify the Integrand:
The integrand is [tex]\(\sec^2 x \cdot \csc^2 x\)[/tex]. We know the definitions of [tex]\(\sec x\)[/tex] and [tex]\(\csc x\)[/tex]:
[tex]\[
\sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x}
\][/tex]
Therefore,
[tex]\[
\sec^2 x = \frac{1}{\cos^2 x} \quad \text{and} \quad \csc^2 x = \frac{1}{\sin^2 x}
\][/tex]
2. Rewrite the Integrand:
Using these definitions, we can rewrite [tex]\(\sec^2 x \cdot \csc^2 x\)[/tex] as follows:
[tex]\[
\sec^2 x \cdot \csc^2 x = \left(\frac{1}{\cos^2 x}\right) \cdot \left(\frac{1}{\sin^2 x}\right) = \frac{1}{\cos^2 x \cdot \sin^2 x}
\][/tex]
3. Simplify using a Trigonometric Identity:
Recall that [tex]\(\cos^2 x \sin^2 x\)[/tex] is related to [tex]\(\sin 2x\)[/tex]:
[tex]\[
\cos(2x) = \cos^2 x - \sin^2 x \quad \text{and} \quad \sin(2x) = 2 \sin x \cos x
\][/tex]
Hence,
[tex]\[
\sin^2 x \cos^2 x = \left(\frac{\sin 2x}{2}\right)^2 = \frac{\sin^2 2x}{4}
\][/tex]
4. Substitute and Integrate:
Substitute this back into the integrand:
[tex]\[
\sec^2 x \cdot \csc^2 x = \frac{1}{\frac{\sin^2 2x}{4}} = \frac{4}{\sin^2 2x} = 4 \csc^2 2x
\][/tex]
The integral now becomes:
[tex]\[
\int \sec^2 x \cdot \csc^2 x \, dx = \int 4 \csc^2 2x \, dx
\][/tex]
5. Variable Substitution:
Let [tex]\(u = 2x\)[/tex]. Then [tex]\(du = 2dx\)[/tex] or [tex]\(dx = \frac{du}{2}\)[/tex].
Substitute [tex]\(u\)[/tex] into the integral:
[tex]\[
\int 4 \csc^2 2x \, dx = \int 4 \csc^2 u \cdot \frac{du}{2} = 2 \int \csc^2 u \, du
\][/tex]
6. Integrate:
Recall the integral of [tex]\(\csc^2 u\)[/tex]:
[tex]\[
\int \csc^2 u \, du = -\cot u
\][/tex]
Thus,
[tex]\[
2 \int \csc^2 u \, du = 2(-\cot u) = -2 \cot u
\][/tex]
7. Back-Substitute [tex]\(u = 2x\)[/tex]:
Finally, reverse the substitution:
[tex]\[
-2 \cot(2x)
\][/tex]
Thus, the solution to the integral [tex]\(\int \sec^2 x \cdot \operatorname{cosec}^2 x \, dx\)[/tex] is:
[tex]\[
-2 \cot(2x)
\][/tex]
