Step 1: Simplify the first factor

We know that $\sec^2 \theta - 1 = \tan^2 \theta$, so $1 - \sec^2 \theta = -\tan^2 \theta$.

$1 - \sec^2 \theta = -\tan^2 \theta$

Step 2: Simplify the second and third factors

Using the algebraic identity $(a - b)(a + b) = a^2 - b^2$, we get $(1 - \sin \theta)(1 + \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta$.

$(1 - \sin \theta)(1 + \sin \theta) = \cos^2 \theta$

Step 3: Simplify the fourth factor

Using the Pythagorean identity for cosecant and cotangent, $1 + \cot^2 \theta = \csc^2 \theta$.

$1 + \cot^2 \theta = \csc^2 \theta$

Step 4: Combine all simplified terms

Substitute all parts back into the expression: $(-\tan^2 \theta) \cdot (\cos^2 \theta) \cdot (\csc^2 \theta)$. Express $\tan^2 \theta$ as $\frac{\sin^2 \theta}{\cos^2 \theta}$ and $\csc^2 \theta$ as $\frac{1}{\sin^2 \theta}$.

$-\left(\frac{\sin^2 \theta}{\cos^2 \theta}\right) \cdot (\cos^2 \theta) \cdot \left(\frac{1}{\sin^2 \theta}\right) = -1$