Question
Which identity needs to be used to prove tangent left-parenthesis Start Fraction pi over 2 End Fraction minus x right-parenthesis equals cotangent x?
(1 point)
Responses
Apply tangent left parenthesis alpha minus beta right parenthesis equals start fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction.
Apply Image with alt text: tangent left parenthesis alpha minus beta right parenthesis equals start fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction .
Apply tangent theta equals Start Fraction sine theta over cosine theta End Fraction.
Apply Image with alt text: tangent theta equals Start Fraction sine theta over cosine theta End Fraction .
Apply tangent left-parenthesis negative x right-parenthesis equals negative tangent x.
Apply Image with alt text: tangent left-parenthesis negative x right-parenthesis equals negative tangent x .
Apply cosine left parenthesis alpha minus beta right parenthesis equals cosine alpha cosine beta plus sine alpha sine beta.
(1 point)
Responses
Apply tangent left parenthesis alpha minus beta right parenthesis equals start fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction.
Apply Image with alt text: tangent left parenthesis alpha minus beta right parenthesis equals start fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction .
Apply tangent theta equals Start Fraction sine theta over cosine theta End Fraction.
Apply Image with alt text: tangent theta equals Start Fraction sine theta over cosine theta End Fraction .
Apply tangent left-parenthesis negative x right-parenthesis equals negative tangent x.
Apply Image with alt text: tangent left-parenthesis negative x right-parenthesis equals negative tangent x .
Apply cosine left parenthesis alpha minus beta right parenthesis equals cosine alpha cosine beta plus sine alpha sine beta.
Answers
GPT-4o mini
To prove the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), we need to utilize the properties of the tangent function and particularly the complementary angle identity.
The correct identity to apply here is:
**Apply tangent left-parenthesis alpha minus beta right-parenthesis equals Start Fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction.**
This identity relates the tangent of a difference of angles to the tangents of the individual angles. However, in this specific case, we can directly make use of the fundamental property:
\[
\tan\left(\frac{\pi}{2} - x\right) = \cot x
\]
This stems from the fact that:
\[
\cot x = \frac{1}{\tan x}
\]
Thus, while the direct complementary angle identity \(\tan\left(\frac{\pi}{2} - x\right) = \cot x\) is a specific case, knowing how to derive it involves recognizing that it links to the fundamental definitions of tangent and cotangent being reciprocals of each other.
So while the first option is the broader way to handle other tangent angle identities, for this specific identity, the most relevant is the basic complementary angle identity itself:
\[
\tan\left(\frac{\pi}{2} - x\right) = \cot x
\]
If you have to choose among your provided options focusing on this specific proof, you may only see a mention of the first identity in a broad sense. The relationship holds primarily due to the properties of tangent and cotangent.
The correct identity to apply here is:
**Apply tangent left-parenthesis alpha minus beta right-parenthesis equals Start Fraction tangent alpha minus tangent beta over 1 plus tangent alpha tangent beta end fraction.**
This identity relates the tangent of a difference of angles to the tangents of the individual angles. However, in this specific case, we can directly make use of the fundamental property:
\[
\tan\left(\frac{\pi}{2} - x\right) = \cot x
\]
This stems from the fact that:
\[
\cot x = \frac{1}{\tan x}
\]
Thus, while the direct complementary angle identity \(\tan\left(\frac{\pi}{2} - x\right) = \cot x\) is a specific case, knowing how to derive it involves recognizing that it links to the fundamental definitions of tangent and cotangent being reciprocals of each other.
So while the first option is the broader way to handle other tangent angle identities, for this specific identity, the most relevant is the basic complementary angle identity itself:
\[
\tan\left(\frac{\pi}{2} - x\right) = \cot x
\]
If you have to choose among your provided options focusing on this specific proof, you may only see a mention of the first identity in a broad sense. The relationship holds primarily due to the properties of tangent and cotangent.