Question

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.
Apply Image with alt text: cosine left parenthesis alpha minus beta right parenthesis equals cosine alpha cosine beta plus sine alpha sine beta .
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Answer 1

To prove the identity \(\tan\left(\frac{\pi}{2} - x\right) = \cot x\), you would want to apply the co-function identity for tangent, which states that:

\[ \tan\left(\frac{\pi}{2} - x\right) = \cot x. \]

However, from the options provided, the identity that would actually be useful is the one that starts with the definition of tangent in terms of sine and cosine:

Apply \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Using this definition, you can substitute \(\frac{\pi}{2} - x\) for \(\theta\):

\[ \tan\left(\frac{\pi}{2} - x\right) = \frac{\sin\left(\frac{\pi}{2} - x\right)}{\cos\left(\frac{\pi}{2} - x\right)}. \]

Knowing the sine and cosine of \(\frac{\pi}{2} - x\):

\[ \sin\left(\frac{\pi}{2} - x\right) = \cos x \quad \text{and} \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x, \]

this gives:

\[ \tan\left(\frac{\pi}{2} - x\right) = \frac{\cos x}{\sin x} = \cot x. \]

Thus, the correct choice is:

Apply \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).