Questions: Match the expression with its equivalent. Not all answers will be used. sed(x) 1 / cos (x) cot (x) Choose tan ^ 2(x) [Choose] tan (x) [Choose] csc (x) [Choose] tan ^ 2(x)+1 [Choose]

Solution Steps

To match the given expressions with their equivalents, we need to understand the trigonometric identities and relationships between the functions. Here are the steps:

1. Identify the trigonometric identities for each given function.
2. Match each function with its equivalent expression based on these identities.

Using the identity \( \sec(x) = \frac{1}{\cos(x)} \), we find: \[ \sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \sqrt{2} \]

Using the identity \( \cot(x) = \frac{1}{\tan(x)} \), we find: \[ \cot\left(\frac{\pi}{4}\right) = \frac{1}{\tan\left(\frac{\pi}{4}\right)} = 1 \]

Using the identity \( \tan^2(x) = \tan(x) \cdot \tan(x) \), we find: \[ \tan^2\left(\frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right)^2 = 1 \]

Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we find: \[ \tan\left(\frac{\pi}{4}\right) = 1 \]

Using the identity \( \csc(x) = \frac{1}{\sin(x)} \), we find: \[ \csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \sqrt{2} \]

Using the identity \( \tan^2(x) + 1 = \sec^2(x) \), we find: \[ \tan^2\left(\frac{\pi}{4}\right) + 1 = 1 + 1 = 2 \]

Final Answer