https://file.aiquickdraw.com/mathbot/file/1728820128057xprg0ic0.jpg

To simplify \( \cos(-x) \sin(-x) \sec(-x) \) using fundamental trigonometric identities, let's proceed step by step:

### Step 1: Use the Even-Odd Identities
The trigonometric functions \(\cos(-x)\), \(\sin(-x)\), and \(\sec(-x)\) can be simplified using the following properties:

- \( \cos(-x) = \cos(x) \) (cosine is an even function)
- \( \sin(-x) = -\sin(x) \) (sine is an odd function)
- \( \sec(-x) = \sec(x) \) (secant is an even function, since \( \sec(x) = \frac{1}{\cos(x)} \))

Thus, we can rewrite the expression as:

\[
\cos(-x) \sin(-x) \sec(-x) = \cos(x) (-\sin(x)) \sec(x)
\]

### Step 2: Simplify the Expression
Now substitute \(\sec(x) = \frac{1}{\cos(x)}\) into the expression:

\[
\cos(x) (-\sin(x)) \sec(x) = \cos(x) (-\sin(x)) \frac{1}{\cos(x)}
\]

Cancel out the \( \cos(x) \) terms:

\[
-\sin(x)
\]

### Final Answer:
The fully simplified expression is:

\[
-\sin(x)
\]

Would you like a detailed explanation of any step? Let me know if you have questions!

Here are 5 related questions to expand your understanding:
1. What are even and odd functions, and how do they apply to trigonometric identities?
2. How do the reciprocal identities (like secant, cosecant) simplify expressions involving trigonometric functions?
3. Can you apply the same process to simplify an expression like \( \tan(-x) \sin(-x) \)?
4. Why is \(\sec(-x)\) equal to \(\sec(x)\), even though \(\sin(-x)\) changes sign?
5. How would this process change if you were given a product like \( \cot(-x) \cos(-x) \sec(-x) \)?

**Tip:** Always start with checking if any trigonometric identities (like even-odd properties) can simplify terms before applying other identities.

Use the fundamental identities to fully simplify cos(-x)sin(-x)sec(-x).

Learn how to simplify cos(-x)sin(-x)sec(-x) using fundamental trigonometric identities, such as even-odd and reciprocal identities. The step-by-step solution is suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation