Question

Let \( P = \{ \theta \in [0, 4\pi] : \tan^2\theta \neq 1 \} \) and \( S = \{ a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta) \sec 2\theta = a^2, \theta \in P \} \). Then \( n(S) \) is:

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

Step 1: Understanding the set \( P \).
We are given that \( P = \{ \theta \in [0, 4\pi] : \tan^2\theta \neq 1 \} \). This means that \( \theta \) must not be an odd multiple of \( \frac{\pi}{4} \), where \( \tan^2\theta = 1 \). So, \( \theta \neq \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \).
Step 2: Understanding the set \( S \).
Next, we are given that \( S = \{ a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta) \sec 2\theta = a^2, \theta \in P \} \). We need to analyze the expression \( 2(\cos^8\theta - \sin^8\theta) \sec 2\theta \). Using the identity \( \cos^8\theta - \sin^8\theta = (\cos^4\theta - \sin^4\theta)(\cos^2\theta + \sin^2\theta) \), and simplifying, we find that no integer values of \( a \) satisfy the equation for \( \theta \) values in \( P \).
Step 3: Conclusion.
Thus, the set \( S \) is empty, so the number of elements in \( S \), \( n(S) \), is 0.
Final Answer: (A) 0