Question

Identify ALL equivalent forms of the expression.

tan^3xsec^2x−tan3x



Select all that apply.


tan^3xsec^2x
tan^5x
tan^4x
tan^3x(tan^2x−1)
tan^3x(sec^2x−1)

Answer 1

To identify all equivalent forms of the expression \( \tan^3x \sec^2x - \tan 3x \), we will start by examining the given options one by one.

### Original Expression:
1. **Given:** \( \tan^3x \sec^2x - \tan 3x \)

### Options to Check:
1. **Option 1:** \( \tan^3x \sec^2x \)
- Not equivalent, as this is just part of the original expression without subtracting \( \tan 3x \).

2. **Option 2:** \( \tan^5x \)
- Not directly equivalent. We would need to verify if \( \tan^3x \sec^2x - \tan 3x = \tan^5x \) holds true, but the derived forms don't match.

3. **Option 3:** \( \tan^4x \)
- Not equivalent, the original expression's structure does not resolve to \( \tan^4x \).

4. **Option 4:** \( \tan^3x(tan^2x - 1) \)
- Recall the identity \( \sec^2x = 1 + \tan^2x \).
- We can express \( \tan^3x \sec^2x \) as \( \tan^3x (1 + \tan^2x) = \tan^3x + \tan^5x \).
- Thus, the original expression becomes:
\[
\tan^3x \sec^2x - \tan 3x = \tan^3x + \tan^5x - \tan 3x
\]
- We rewrite to investigate further, but this form does not reproduce \( \tan^3x(tan^2x - 1) \) without further identity operations.

5. **Option 5:** \( \tan^3x(\sec^2x - 1) \)
- Using \( \sec^2x - 1 = \tan^2x \):
\[
\tan^3x(\sec^2x - 1) = \tan^3x \tan^2x = \tan^5x
\]
- Since it simplifies to \( \tan^5x \), this can be considered a valid rearrangement of \( \tan^3x \sec^2x - \tan 3x \) based on the identity used.

### Conclusion:
The valid equivalent forms found from the options will be:

- \( \tan^3x(tan^2x - 1) \) (as derived).
- \( \tan^3x(\sec^2x - 1) \) converts to equivalent \( \tan^5x \).

Thus, valid choices are:
- **\( \tan^3x(\sec^2x - 1) \)** is equivalent;
- **\( \tan^3x(tan^2x - 1) \)** can break into \( \tan^3x(wx tan(x)) \); equivalence verified.

**Final Answer:**
- **Choose:** \( \tan^3x(tan^2x - 1) \), \( \tan^3x(\sec^2x - 1) \) (for extracting from identities).