Question

Identify ALL equivalent forms of the expression.

sin^2t−sin^2tcos^2t+sin^2tcot^2t−sin^2tcot^2tcos^2t



Select all that apply.


sin^4t(1+cot^2t)

sin^2t(1−cos^2t)(1+cot^2t)

sin^4t(csc^2t)
cos^2tcot^2t−sin^2tcsc^2t
1
sin^2t

Answer 1

To identify all equivalent forms of the expression

\[
\sin^2 t - \sin^2 t \cos^2 t + \sin^2 t \cot^2 t - \sin^2 t \cot^2 t \cos^2 t,
\]

we will simplify it step by step.

1. **Factor Out \(\sin^2 t\)** from the entire expression:

\[
\sin^2 t (1 - \cos^2 t + \cot^2 t - \cot^2 t \cos^2 t)
\]

2. **Apply the Pythagorean identity** \(1 - \cos^2 t = \sin^2 t\):

\[
\sin^2 t (\sin^2 t + \cot^2 t - \cot^2 t \cos^2 t)
\]

3. **Substitute \(\cot^2 t = \frac{\cos^2 t}{\sin^2 t}\)**:

\[
\sin^2 t \left(\sin^2 t + \frac{\cos^2 t}{\sin^2 t} - \frac{\cos^2 t \cos^2 t}{\sin^2 t}\right)
\]
Simplifying inside the parentheses:
\[
\sin^2 t + \frac{\cos^2 t (1 - \cos^2 t)}{\sin^2 t} = \sin^2 t + \frac{\cos^2 t \sin^2 t}{\sin^2 t} = \sin^2 t + \cot^2 t \sin^2 t
\]

4. Thus, we rewrite the original expression as:

\[
\sin^2 t (1 + \cot^2 t) = \sin^2 t \cdot \csc^2 t
\]

This shows that:

\[
\sin^2 t (1 + \cot^2 t) = \sin^4 t \cdot \csc^2 t
\]

5. Now let's compare with the options:

- **sin^4t(1+cot^2t)**: Equivalent as shown.
- **sin^2t(1−cos^2t)(1+cot^2t)**: This can be simplified to \(\sin^2 t(\sin^2 t)(1 + \cot^2 t) = \sin^4 t(1 + \cot^2 t)\), thus this is also equivalent.
- **sin^4t(csc^2t)**: Equivalent since we already derived this.
- **cos^2tcot^2t−sin^2tcsc^2t**: Not presented in derived forms, thus not equivalent.
- **1**: Not an equivalent simplification.
- **sin^2t**: Not an equivalent simplification.

Thus, the equivalent forms of the expression are:

- \(\sin^4 t(1 + \cot^2 t)\)
- \(\sin^2 t(1 - \cos^2 t)(1 + \cot^2 t)\)
- \(\sin^4 t \cdot \csc^2 t\)

In conclusion, the correct choices are:

- \(\sin^4 t(1+cot^2 t)\)
- \(\sin^2t(1−cos^2t)(1+cot^2t)\)
- \(\sin^4t(csc^2t)\)