Question

The expression \( \frac{\sec^6 \theta - \tan^6 \theta - 3 \sec^2 \theta \tan^2 \theta}{1 + 2 \sin^2 \theta - \sin^4 \theta + \cos^4 \theta} \) is equal to:

• A. \( 1 \)
• B. \( \frac{1}{2} \)
• C. \( 2 \)
• D. \( \frac{1}{4} \)

Hint:

For trigonometric simplifications, look for common identities such as \( \sec^2 \theta - \tan^2 \theta = 1 \), \( \sin^2 \theta + \cos^2 \theta = 1 \), and others that help reduce complex expressions.

Answer 1

The Correct Option is B

We are given the expression: \[ \frac{\sec^6 \theta - \tan^6 \theta - 3 \sec^2 \theta \tan^2 \theta}{1 + 2 \sin^2 \theta - \sin^4 \theta + \cos^4 \theta} \] We can simplify both the numerator and denominator step by step. 1. Numerator: The numerator is \( \sec^6 \theta - \tan^6 \theta - 3 \sec^2 \theta \tan^2 \theta \). Use the identity \( \sec^2 \theta - \tan^2 \theta = 1 \) and simplify. 2. Denominator: The denominator is \( 1 + 2 \sin^2 \theta - \sin^4 \theta + \cos^4 \theta \). Factor the trigonometric expressions and simplify the terms. After simplifying both the numerator and the denominator, the expression reduces to \( \frac{1}{2} \). Thus, the correct answer is \( \frac{1}{2} \).