Questions: Determine if the following statement is true or false. The graphs of y=sec(x/2) and y=cos(x/2) are identical. Choose the correct choice below. A. False. The cosine graph has a range of (-∞,-1] ∪[1, ∞), and the secant graph has a range of [-1,1]. B. True. Both secant and cosine graph's have a range of (-∞,-1] ∪[1, ∞). C. False. The secant graph has a range of (-∞,-1] ∪[1, ∞), and the cosine graph has a range of [-1,1]. D. True. Both secant and cosine graphs have vertical asymptotes at interval multiples of π.

Solution Steps

To determine if the graphs of \( y = \sec \frac{x}{2} \) and \( y = \cos \frac{x}{2} \) are identical, we need to analyze the properties of the secant and cosine functions. Specifically, we should compare their ranges and identify any vertical asymptotes.

1. The range of the cosine function \( y = \cos \frac{x}{2} \) is \([-1, 1]\).
2. The range of the secant function \( y = \sec \frac{x}{2} \) is \((-\infty, -1] \cup [1, \infty)\).
3. The secant function has vertical asymptotes where the cosine function is zero, which occur at \( x = (2k+1)\pi \) for integer \( k \).

Based on these properties, we can determine the correct choice.

The cosine function \( y = \cos \frac{x}{2} \) has a range of \( [-1, 1] \). This means that the output values of the cosine function will always lie between -1 and 1, inclusive.

The secant function \( y = \sec \frac{x}{2} \) is defined as the reciprocal of the cosine function. Therefore, its range is \( (-\infty, -1] \cup [1, \infty) \). This indicates that the secant function can take values less than or equal to -1 and greater than or equal to 1, but it cannot take any values between -1 and 1.

From the analysis:

• The range of \( y = \cos \frac{x}{2} \) is \( [-1, 1] \).
• The range of \( y = \sec \frac{x}{2} \) is \( (-\infty, -1] \cup [1, \infty) \).

Since the ranges are different, the graphs of \( y = \sec \frac{x}{2} \) and \( y = \cos \frac{x}{2} \) cannot be identical.

• Option A: False. The cosine graph has a range of \( (-\infty,-1] \cup[1, \infty) \), and the secant graph has a range of \( [-1,1] \). (Incorrect)
• Option B: True. Both secant and cosine graphs have a range of \( (-\infty,-1] \cup[1, \infty) \). (Incorrect)
• Option C: False. The secant graph has a range of \( (-\infty,-1] \cup[1, \infty) \), and the cosine graph has a range of \( [-1,1] \). (Correct)
• Option D: True. Both secant and cosine graphs have vertical asymptotes at interval multiples of \( \pi \). (Incorrect)

Final Answer