Question
Which are simplified forms of the expression secant squared theta sine 2 theta? Select all that apply.
(1 point)
Responses
2 cot θ
2 cot θ
2 minus secant squared theta
Image with alt text: 2 minus secant squared theta
2 tan θ
2 tan θ
secant squared theta minus 1
Image with alt text: secant squared theta minus 1
Start Fraction sine 2 theta over cosine squared theta End Fraction
(1 point)
Responses
2 cot θ
2 cot θ
2 minus secant squared theta
Image with alt text: 2 minus secant squared theta
2 tan θ
2 tan θ
secant squared theta minus 1
Image with alt text: secant squared theta minus 1
Start Fraction sine 2 theta over cosine squared theta End Fraction
Answers
GPT-4o mini
To simplify the expression \( \sec^2 \theta \sin 2\theta \), we can use the definitions and identities for secant and sine.
First, recall the identities:
1. \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \)
2. \( \sin 2\theta = 2 \sin \theta \cos \theta \)
Substituting these into the expression gives us:
\[
\sec^2 \theta \sin 2\theta = \left(\frac{1}{\cos^2 \theta}\right) (2 \sin \theta \cos \theta)
\]
This simplifies to:
\[
= \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} = 2 \frac{\sin \theta}{\cos \theta} = 2 \tan \theta
\]
So, \( \sec^2 \theta \sin 2\theta \) simplifies to \( 2 \tan \theta \).
Now let's evaluate the given options to see which ones are equivalent to \( 2 \tan \theta \):
1. **\( 2 \cot \theta \)**: This is not equivalent to \( 2 \tan \theta \).
2. **\( 2 - \sec^2 \theta \)**: This does not simplify to \( 2 \tan \theta \) either.
3. **\( 2 \tan \theta \)**: This is the simplified form we derived, so it is a valid response.
4. **\( \sec^2 \theta - 1 \)**: This is equal to \( \tan^2 \theta \), which is not equivalent to \( 2 \tan \theta \).
5. **Start Fraction sine 2 theta over cosine squared theta End Fraction**: This is equal to \( \sec^2 \theta \sin 2\theta\) as we derived in the first step.
Thus, the only response that represents a simplified form of \( \sec^2 \theta \sin 2\theta\) is:
- **\( 2 \tan \theta \)**
- **Start Fraction sine 2 theta over cosine squared theta End Fraction**
To conclude, the valid selections are:
- \( 2 \tan \theta \)
- Start Fraction sine 2 theta over cosine squared theta End Fraction
First, recall the identities:
1. \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \)
2. \( \sin 2\theta = 2 \sin \theta \cos \theta \)
Substituting these into the expression gives us:
\[
\sec^2 \theta \sin 2\theta = \left(\frac{1}{\cos^2 \theta}\right) (2 \sin \theta \cos \theta)
\]
This simplifies to:
\[
= \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} = 2 \frac{\sin \theta}{\cos \theta} = 2 \tan \theta
\]
So, \( \sec^2 \theta \sin 2\theta \) simplifies to \( 2 \tan \theta \).
Now let's evaluate the given options to see which ones are equivalent to \( 2 \tan \theta \):
1. **\( 2 \cot \theta \)**: This is not equivalent to \( 2 \tan \theta \).
2. **\( 2 - \sec^2 \theta \)**: This does not simplify to \( 2 \tan \theta \) either.
3. **\( 2 \tan \theta \)**: This is the simplified form we derived, so it is a valid response.
4. **\( \sec^2 \theta - 1 \)**: This is equal to \( \tan^2 \theta \), which is not equivalent to \( 2 \tan \theta \).
5. **Start Fraction sine 2 theta over cosine squared theta End Fraction**: This is equal to \( \sec^2 \theta \sin 2\theta\) as we derived in the first step.
Thus, the only response that represents a simplified form of \( \sec^2 \theta \sin 2\theta\) is:
- **\( 2 \tan \theta \)**
- **Start Fraction sine 2 theta over cosine squared theta End Fraction**
To conclude, the valid selections are:
- \( 2 \tan \theta \)
- Start Fraction sine 2 theta over cosine squared theta End Fraction