https://file.aiquickdraw.com/mathbot/file/17306176308130sa76day.jpg

The problem asks to express the tangent function \( \tan(\theta) \) in terms of the secant function \( \sec(\theta) \), and it offers several multiple-choice answers.

### Step-by-Step Solution
We know from trigonometric identities that:
\[
\tan(\theta) = \pm \sqrt{\sec^2(\theta) - 1}
\]

This identity comes from the Pythagorean identity:
\[
\tan^2(\theta) + 1 = \sec^2(\theta)
\]
Rearranging, we get:
\[
\tan^2(\theta) = \sec^2(\theta) - 1
\]
Taking the square root on both sides gives:
\[
\tan(\theta) = \pm \sqrt{\sec^2(\theta) - 1}
\]

### Conclusion
The correct answer is therefore:
\[
\boxed{\pm \sqrt{\sec^2(\theta) - 1}}
\]
This corresponds to option **B** in the choices provided.

Would you like further explanation on how the identity was derived or details about tangent and secant relationships?

---

### Related Questions
1. What is the relationship between \(\sec(\theta)\) and \(\cos(\theta)\)?
2. How can we express \(\sin(\theta)\) in terms of \(\sec(\theta)\)?
3. What are other Pythagorean identities involving trigonometric functions?
4. How do we handle the \(\pm\) sign in trigonometric identities?
5. What is the definition of the tangent function in terms of sine and cosine?

### Tip
Remember that the square root introduces a \(\pm\) symbol, so it's important to consider the quadrant where \(\theta\) is located to determine the correct sign in trigonometric functions.

Write the first trigonometric function tan(θ) in terms of the second trigonometric function sec(θ).

Solve for tan(θ) in terms of sec(θ) using the Pythagorean identity. This problem provides a step-by-step solution, highlighting key trigonometric concepts for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation