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Which of the following is \( \cot (\theta) \sec (\theta) \) in simplified form? \( \csc \theta \) \( \tan \theta \) -1 1

Ask by May Park. in the United States
Mar 11,2025

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The simplified form of \( \cot (\theta) \sec (\theta) \) is \( \csc \theta \).

Solución

We start by recalling the definitions: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta}. \] Now, multiply the two expressions: \[ \cot \theta \sec \theta = \left(\frac{\cos \theta}{\sin \theta}\right) \left(\frac{1}{\cos \theta}\right). \] The \(\cos \theta\) terms cancel: \[ \cot \theta \sec \theta = \frac{1}{\sin \theta} = \csc \theta. \] Thus, the simplified form is \(\csc \theta\).

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To simplify \( \cot (\theta) \sec (\theta) \), we can express the trigonometric functions in terms of sine and cosine. Recall that \( \cot (\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \sec (\theta) = \frac{1}{\cos(\theta)} \). Therefore, \( \cot (\theta) \sec (\theta) = \frac{\cos(\theta)}{\sin(\theta)} \cdot \frac{1}{\cos(\theta)} = \frac{1}{\sin(\theta)} = \csc(\theta) \). So, the simplified expression is \( \csc(\theta) \). For a fun twist, remember that the cotangent and secant functions always have a party together on the unit circle! Cotangent dresses up in cosine while secant brings its vibrant 1/cosine style. When they combine their efforts, they transform into their sine-celebrating friend \( \csc(\theta) \)! So next time you're calculating, think of it as a trigonometric masquerade!

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