Simplify Trigonometric Expression Using Identities

This solution provides a step-by-step guide to simplifying the given trigonometric expression: \( (1 - \sec^2\theta)(1 - \sin\theta)(1 + \sin\theta)(1 + \cot^2\theta) \)

Trigonometric Identities Review

To simplify the expression, we will use fundamental trigonometric identities:

• Pythagorean Identity related to cotangent: \( 1 + \cot^2\theta = \csc^2\theta \)
• Pythagorean Identity related to tangent: \( \sec^2\theta - \tan^2\theta = 1 \), which implies \( 1 - \sec^2\theta = -\tan^2\theta \)
• Difference of Squares: \( (a - b)(a + b) = a^2 - b^2 \). Applying this to \( (1 - \sin\theta)(1 + \sin\theta) \), we get \( 1 - \sin^2\theta \).
• Co-function Identity: \( 1 - \sin^2\theta = \cos^2\theta \)

Simplification Steps

Let's break down the simplification process:

1. Substitute known identities: Replace parts of the expression with their equivalent forms using the identities.
   • Replace \( 1 + \cot^2\theta \) with \( \csc^2\theta \).
   • Replace \( 1 - \sec^2\theta \) with \( -\tan^2\theta \).
   • Replace \( (1 - \sin\theta)(1 + \sin\theta) \) with \( 1 - \sin^2\theta \), which simplifies to \( \cos^2\theta \).
   The expression becomes: \( (-\tan^2\theta)(\cos^2\theta)(\csc^2\theta) \)
2. Convert to basic trigonometric functions (sine and cosine): Express \( \tan\theta \) and \( \csc\theta \) in terms of \( \sin\theta \) and \( \cos\theta \).
   • \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), so \( \tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta} \)
   • \( \csc\theta = \frac{1}{\sin\theta} \), so \( \csc^2\theta = \frac{1}{\sin^2\theta} \)
   Substitute these into the expression: \( \left(-\frac{\sin^2\theta}{\cos^2\theta}\right)(\cos^2\theta)\left(\frac{1}{\sin^2\theta}\right) \)
3. Cancel out terms: Simplify the expression by cancelling common factors in the numerator and denominator. \( \left(-\frac{\cancel{\sin^2\theta}}{\cancel{\cos^2\theta}}\right) \times \cancel{\cos^2\theta} \times \left(\frac{1}{\cancel{\sin^2\theta}}\right) \) After cancellation, the expression simplifies to \( -1 \).

Final Result

The simplified value of the expression \( (1 - \sec^2\theta)(1 - \sin\theta)(1 + \sin\theta)(1 + \cot^2\theta) \) is -1.