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Determine la validez de las siguientes identidades:
a.
sec (θ)
csc (θ)
− tan (θ) + 3 csc2
(θ) − 3 cot2
(θ) = 3
b. sen (x) =
p sec2
(x) − 1
sec (x)

Let's analyze and determine the validity of the given trigonometric identities. I'll go through each one step by step.

### Identity a:
\[
\frac{\sec(\theta)}{\csc(\theta)} - \tan(\theta) + 3\csc^2(\theta) - 3\cot^2(\theta) = 3
\]

### Steps:
1. **Simplify each term separately**:
   - \(\frac{\sec(\theta)}{\csc(\theta)}\): 
     - Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
     - So, \(\frac{\sec(\theta)}{\csc(\theta)} = \frac{\frac{1}{\cos(\theta)}}{\frac{1}{\sin(\theta)}} = \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)\).

   - Now the identity becomes:
     \[
     \tan(\theta) - \tan(\theta) + 3\csc^2(\theta) - 3\cot^2(\theta) = 3
     \]
     The \(\tan(\theta)\) terms cancel out:
     \[
     3\csc^2(\theta) - 3\cot^2(\theta) = 3
     \]

2. **Factor out the common factor of 3**:
   \[
   3(\csc^2(\theta) - \cot^2(\theta)) = 3
   \]

3. **Use the Pythagorean identity**: 
   - The identity \(\csc^2(\theta) - \cot^2(\theta) = 1\).
   - Therefore, the equation simplifies to:
     \[
     3 \times 1 = 3
     \]

Since both sides of the equation are equal, **this identity is valid**.

### Identity b:
\[
\sin(x) = \frac{\sqrt{\sec^2(x) - 1}}{\sec(x)}
\]

### Steps:
1. **Simplify the expression under the square root**:
   - Recall the identity \(\sec^2(x) = 1 + \tan^2(x)\).
   - So, \(\sec^2(x) - 1 = \tan^2(x)\).
   - Therefore, the equation becomes:
     \[
     \sin(x) = \frac{\sqrt{\tan^2(x)}}{\sec(x)}
     \]
     Simplifying further:
     \[
     \sin(x) = \frac{|\tan(x)|}{\sec(x)}
     \]

2. **Rewrite using basic trigonometric identities**:
   - Recall that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\).
   - Substituting these into the equation:
     \[
     \sin(x) = \frac{\frac{\sin(x)}{\cos(x)}}{\frac{1}{\cos(x)}} = \sin(x)
     \]

Thus, the equation simplifies to \(\sin(x) = \sin(x)\), proving that **this identity is also valid**.

---

Would you like more details on any part of the solution? Here are some related questions to explore further:

1. What are the key trigonometric identities used in simplifying complex expressions?
2. How do you approach verifying more complicated trigonometric identities?
3. Can you explain the Pythagorean identities in more detail?
4. What are the differences between verifying an identity and solving an equation?
5. How can trigonometric identities be applied in solving real-world problems?

**Tip**: Always start by simplifying individual terms when working with identities, and recall basic identities like the Pythagorean identities to help with simplification.

Learn how to verify trigonometric identities involving secant, cosecant, tangent, and cotangent. Step-by-step analysis and simplification are provided to validate the given identities. Suitable for advanced high school students and those preparing for trigonometry exams.

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