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
1. 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 .
2. 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 .
3. 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 .
4. Apply cosine left parenthesis alpha minus beta right parenthesis equals cosine alpha cosine beta plus sine alpha sine beta.
(1 point)
Responses
1. 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 .
2. 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 .
3. 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 .
4. 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 \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), the best identity to use is:
**1. Apply \( \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \).**
This identity will allow you to express \( \tan\left(\frac{\pi}{2} - x\right) \) in terms of known trigonometric values. Specifically, you can set \( \alpha = \frac{\pi}{2} \) and \( \beta = x \), leading to an expression that relates to other trigonometric functions.
However, you could alternatively directly use the definition and properties of tangent and cotangent to prove the identity as well. Keep in mind that using the definition of tangent, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), is useful for understanding the components involved, but for this particular problem the difference angle identity is most directly applicable.
Thus, although using \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \) (option 2) can also lead you to the proof, option 1 is more straightforward for proving \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \).
**1. Apply \( \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \).**
This identity will allow you to express \( \tan\left(\frac{\pi}{2} - x\right) \) in terms of known trigonometric values. Specifically, you can set \( \alpha = \frac{\pi}{2} \) and \( \beta = x \), leading to an expression that relates to other trigonometric functions.
However, you could alternatively directly use the definition and properties of tangent and cotangent to prove the identity as well. Keep in mind that using the definition of tangent, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), is useful for understanding the components involved, but for this particular problem the difference angle identity is most directly applicable.
Thus, although using \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \) (option 2) can also lead you to the proof, option 1 is more straightforward for proving \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \).