The given function is \( \sec ^{2} x \cdot \operatorname{cosec}^{2} x \) and we need to compute the integral of this function with respect to \( x \), i.e., compute \( \int \sec^{2} x \cdot \csc^{2} x \, dx \). [ \( \operatorname{cosec} = \csc \)].
Solution:
\( \begin{aligned} & \int \sec^{2} x \cdot \csc^{2} x \, dx \\ &= \int \frac{1}{\cos^{2} x \cdot \sin^{2} x} \, dx, \quad \because \sec^{2} x = \frac{1}{\cos^{2} x}, \csc^{2} x = \frac{1}{\sin^{2} x} \\ &= \int \frac{1}{\sin^{2} x \cdot \cos^{2} x} \, dx \\ &= \int \frac{4}{4 \sin^{2} x \cdot \cos^{2} x}\, dx \\ &= \int \frac{4}{(2 \sin x \cdot \cos x)^{2}} \, dx \\ &= \int \frac{4}{\sin^{2} 2x}\, dx, \quad \because \sin(2x) = 2\sin(x)\cos(x) \\ &= \int 4 \csc^{2} 2x \, dx \\ \end{aligned} \)Let \( u = 2x \), then \( du = 2 dx \) or \( \displaystyle dx = \frac{du}{2} \). Substitute:
\( \begin{aligned} & \int 4 \csc^{2} 2x \, dx = 4 \int \csc^{2} u \cdot \frac{du}{2} \\ &= 2 \int \csc^{2} u \, du = -2 \cot u + C, \quad \text{C is a constant} \\ &= -2 \cot 2x + C \end{aligned} \)Thus, the integral of the given function is
\( \boxed{\int \sec^{2} x \cdot \csc^{2} x \, dx = -2 \cot 2x + C \\} \)Please let me know in the comments if you find any errors in this solution.