Question
Simplify. \( \cos \left(\frac{\pi}{2}-x\right) \csc \left(\frac{\pi}{2}-x\right) \) \( [?] \)
Ask by Malone Craig. in the United States
Mar 11,2025
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Tutor-Verified Answer
Answer
The simplified expression is \( \tan(x) \).
Solution
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Mind Expander
To simplify \( \cos \left(\frac{\pi}{2}-x\right) \csc \left(\frac{\pi}{2}-x\right) \), we can use the co-function identities. We know that \( \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \) and \( \csc\left(\frac{\pi}{2} - x\right) = \sec(x) \). So, \[ \cos \left(\frac{\pi}{2}-x\right) \csc \left(\frac{\pi}{2}-x\right) = \sin(x) \cdot \sec(x) = \sin(x) \cdot \frac{1}{\cos(x)} = \frac{\sin(x)}{\cos(x)} = \tan(x). \] Thus, the simplified expression is: \[ \tan(x). \]
