For which values of $\theta$ is the expression ${\sec }^2\theta \cos \left(2\theta \right)$ defined on the unit circle?

• A. For all $\theta$ such that $\cos \left(\theta \right)\ne 1$
• B. For all $\theta$ such that $\cos \left(\theta \right)\ne 0$ and $\cos \left(2\theta \right)\ne 0$
• C. For all $\theta$ such that $\sin \left(\theta \right)\ne 0$
• D. For all real values of $\theta$

Answer: (B)

Verified step by step guidance

1. Identify the given expression: \(\sec^{2}(\theta) \cdot \cos(2\theta)\). We want to find for which values of \(\theta\) this expression is defined on the unit circle.
2. Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\). Therefore, \(\sec^{2}(\theta) = \frac{1}{\cos^{2}(\theta)}\). This means the expression is undefined wherever \(\cos(\theta) = 0\) because division by zero is undefined.
3. Next, consider the factor \(\cos(2\theta)\). Since cosine is defined for all real numbers, \(\cos(2\theta)\) itself does not cause any undefined values. However, if \(\cos(2\theta) = 0\), the product will be zero, which is defined, so no restriction arises here.
4. Summarize the domain restrictions: The expression is undefined when \(\cos(\theta) = 0\). Therefore, the expression is defined for all \(\theta\) such that \(\cos(\theta) \neq 0\).
5. Conclude that the expression is defined for all \(\theta\) on the unit circle except where \(\cos(\theta) = 0\), which corresponds to \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.