Question
Prove cosecant left-parenthesis Start Fraction pi over 2 End Fraction minus x right-parenthesis equals secant x.
(1 point)
Responses
1. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x plus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x plus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
2. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x minus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x minus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
3. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x plus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x plus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
4. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x minus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
(1 point)
Responses
1. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x plus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x plus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
2. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x minus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x minus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
3. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x plus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
Image with alt text: cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction sine x plus cosine StartFraction pi over 2 EndFraction cosine x EndFraction equals secant x
4. cosecant left-parenthesis StartFraction pi over 2 EndFraction minus x right-parenthesis equals StartFraction 1 over sine StartFraction pi over 2 EndFraction cosine x minus cosine StartFraction pi over 2 EndFraction sine x EndFraction equals secant x
Answers
GPT-4o mini
To prove the identity \( \csc\left(\frac{\pi}{2} - x\right) = \sec x \), we will use the definitions of cosecant and secant in terms of sine and cosine:
1. The cosecant function is defined as:
\[
\csc x = \frac{1}{\sin x}
\]
Therefore,
\[
\csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin\left(\frac{\pi}{2} - x\right)}
\]
2. Using the co-function identity, we have:
\[
\sin\left(\frac{\pi}{2} - x\right) = \cos x
\]
3. Thus, we can write:
\[
\csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\cos x}
\]
4. The secant function is defined as:
\[
\sec x = \frac{1}{\cos x}
\]
5. Therefore, we have:
\[
\csc\left(\frac{\pi}{2} - x\right) = \sec x
\]
This proves that:
\[
\csc\left(\frac{\pi}{2} - x\right) = \sec x
\]
Now, looking through the options given, we want to find which one correctly represents this proof.
The correct response is:
1. \( \csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin\left(\frac{\pi}{2}\right)\cos x + \cos\left(\frac{\pi}{2}\right)\sin x} = \sec x \)
Since \( \sin\left(\frac{\pi}{2}\right) = 1 \) and \( \cos\left(\frac{\pi}{2}\right) = 0 \), this simplifies to \( \frac{1}{\cos x} = \sec x \).
Therefore, the correct option is **1**.
1. The cosecant function is defined as:
\[
\csc x = \frac{1}{\sin x}
\]
Therefore,
\[
\csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin\left(\frac{\pi}{2} - x\right)}
\]
2. Using the co-function identity, we have:
\[
\sin\left(\frac{\pi}{2} - x\right) = \cos x
\]
3. Thus, we can write:
\[
\csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\cos x}
\]
4. The secant function is defined as:
\[
\sec x = \frac{1}{\cos x}
\]
5. Therefore, we have:
\[
\csc\left(\frac{\pi}{2} - x\right) = \sec x
\]
This proves that:
\[
\csc\left(\frac{\pi}{2} - x\right) = \sec x
\]
Now, looking through the options given, we want to find which one correctly represents this proof.
The correct response is:
1. \( \csc\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin\left(\frac{\pi}{2}\right)\cos x + \cos\left(\frac{\pi}{2}\right)\sin x} = \sec x \)
Since \( \sin\left(\frac{\pi}{2}\right) = 1 \) and \( \cos\left(\frac{\pi}{2}\right) = 0 \), this simplifies to \( \frac{1}{\cos x} = \sec x \).
Therefore, the correct option is **1**.