Which of the following options is equivalent to the given expression?
 
\[\dfrac{\sec(x) - 1}{\sin^2(x)+\cos^2(x)}\]

We are given the expression:

\[
\dfrac{\sec(x) - 1}{\sin^2(x) + \cos^2(x)}
\]

Let's simplify this expression step by step.

1. **Simplify the denominator**: 
   The denominator is \(\sin^2(x) + \cos^2(x)\). According to the Pythagorean identity, we know that:

   \[
   \sin^2(x) + \cos^2(x) = 1
   \]

   So, the expression simplifies to:

   \[
   \dfrac{\sec(x) - 1}{1}
   \]

2. **Final expression**:
   Since the denominator is now 1, the expression simplifies to:

   \[
   \sec(x) - 1
   \]

Thus, the expression is equivalent to \(\sec(x) - 1\).

Would you like further details or have any other questions?

Here are some related questions:
1. What is the Pythagorean identity in trigonometry?
2. How do you simplify expressions involving secant and cosecant?
3. What are some common trigonometric identities used in simplifications?
4. How does the secant function relate to the cosine function?
5. Can we simplify \(\sec(x) - 1\) further if given additional trigonometric identities?

**Tip:** Remember that simplifying trigonometric expressions often involves applying known identities such as the Pythagorean identity, angle sum/difference identities, or co-function identities.

Learn how to simplify the expression \(\frac{\sec(x) - 1}{\sin^2(x) + \cos^2(x)}\) using trigonometric identities, specifically the Pythagorean identity. Understand the steps involved and gain insights into trigonometric simplification. Suitable for high school students and those studying trigonometry.

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