Step-by-step Solution:

To evaluate the expression: \[ \left[ \tan^2 (90 - \theta) - \sin^2 (90 - \theta) \right] \csc^2 (90 - \theta) \cot^2 (90 - \theta), \] we can use trigonometric identities to simplify it. --- Step 1: Simplify the Angles Recall that: \[ \tan(90 - \theta) = \cot \theta, \quad \sin(90 - \theta) = \cos \theta, \quad \csc(90 - \theta) = \sec \theta, \quad \cot(90 - \theta) = \tan \theta. \] Substitute these into the expression: \[ \left[ \cot^2 \theta - \cos^2 \theta \right] \sec^2 \theta \tan^2 \theta. \] --- Step 2: Simplify the Expression Rewrite \( \sec^2 \theta \) and \( \tan^2 \theta \) using their identities: \[ \sec^2 \theta = 1 + \tan^2 \theta, \quad \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}. \] Substitute these into the expression: \[ \left[ \cot^2 \theta - \cos^2 \theta \right] (1 + \tan^2 \theta) \tan^2 \theta. \] --- Step 3: Further Simplification Express \( \cot^2 \theta \) as \( \frac{\cos^2 \theta}{\sin^2 \theta} \): \[ \left[ \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta \right] (1 + \tan^2 \theta) \tan^2 \theta. \] Factor out \( \cos^2 \theta \): \[ \cos^2 \theta \left[ \frac{1}{\sin^2 \theta} - 1 \right] (1 + \tan^2 \theta) \tan^2 \theta. \] Simplify \( \frac{1}{\sin^2 \theta} - 1 \): \[ \frac{1 - \sin^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta}{\sin^2 \theta}. \] Now the expression becomes: \[ \cos^2 \theta \cdot \frac{\cos^2 \theta}{\sin^2 \theta} \cdot (1 + \tan^2 \theta) \cdot \tan^2 \theta. \] --- Step 4: Combine Terms Combine the terms: \[ \frac{\cos^4 \theta}{\sin^2 \theta} \cdot (1 + \tan^2 \theta) \cdot \tan^2 \theta. \] Substitute \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \): \[ \frac{\cos^4 \theta}{\sin^2 \theta} \cdot \left(1 + \frac{\sin^2 \theta}{\cos^2 \theta}\right) \cdot \frac{\sin^2 \theta}{\cos^2 \theta}. \] Simplify \( 1 + \frac{\sin^2 \theta}{\cos^2 \theta} \): \[ \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}. \] Now the expression becomes: \[ \frac{\cos^4 \theta}{\sin^2 \theta} \cdot \frac{1}{\cos^2 \theta} \cdot \frac{\sin^2 \theta}{\cos^2 \theta}. \] Simplify: \[ \frac{\cos^4 \theta}{\sin^2 \theta} \cdot \frac{\sin^2 \theta}{\cos^4 \theta} = 1. \] --- Step 5: Conclusion The value of the expression is: \[ \boxed{1} \] Correct Answer: \(\boxed{B}\)