Question

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

Hint:

Always try to use standard identities first. Here, \( \cos^2\theta - \sin^2\theta = \cos 2\theta \), which makes the whole expression very simple.

Answer 1

Correct Answer: 2

Step 1: Simplify the given expression.
We know that
\[ \cos^2\theta - \sin^2\theta = \cos 2\theta \] So, the given expression becomes
\[ (\cos^2\theta - \sin^2\theta)\sec 2\theta = \cos 2\theta \cdot \sec 2\theta \] Since \( \sec 2\theta = \dfrac{1}{\cos 2\theta} \), we get
\[ \cos 2\theta \cdot \sec 2\theta = 1 \]
Step 2: Use the condition on \( \theta \).
The condition \( \tan^2\theta \neq 1 \) means
\[ \tan\theta \neq \pm 1 \] This happens when
\[ \theta \neq \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, \frac{13\pi}{4}, \frac{15\pi}{4} \] At these values, \( \cos 2\theta = 0 \), so \( \sec 2\theta \) is not defined. These are already excluded in set \( P \).
Hence, for every valid \( \theta \in P \), the expression is always equal to
\[ 1 \]
Step 3: Compare with \( a^2 \).
Now the set definition gives
\[ a^2 = 1 \] Since \( a \in \mathbb{Z} \), the possible integer values of \( a \) are
\[ a = 1 \quad \text{or} \quad a = -1 \]
Step 4: Find the number of elements in \( S \).
Therefore,
\[ S = \{-1, 1\} \] So,
\[ n(S) = 2 \]