Which of the following is equivalent to sec ((pi)/(5)) ? sec ((14 pi)/(5)) sec(-(pi)/(5)) sec ((6pi)/(5)) sec ((4pi)/(5))

Question

Which of the following is equivalent to  sec ((pi)/(5)) ?  sec ((14 pi)/(5))  sec(-(pi)/(5))  sec ((6pi)/(5))  sec ((4pi)/(5))

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Properties of Secant Function: Understand the properties of the secant function. The secant function, sec(θ), is periodic with a period of 2π. This means that sec(θ) = sec(θ + 2πk) for any integer k. Additionally, sec(-θ) = sec(θ) because secant is an even function.Evaluate First Option: Evaluate the first option, sec((14π)/(5)). Since the secant function has a period of 2π, we can subtract 2π from (14π)/(5) until we are within the interval [0, 2π). (14π)/(5) - 2π = (14π)/(5) - (10π)/(5) = (4π)/(5). So, sec((14π)/(5)) is equivalent to sec((4π)/(5)).Evaluate Second Option: Evaluate the second option, sec(-(π)/(5)). Since the secant function is even, sec(-(π)/(5)) is equivalent to sec((π)/(5)).Evaluate Third Option: Evaluate the third option, sec((6π)/(5)). Subtracting 2π from (6π)/(5) gives us (6π)/(5) - 2π = (6π)/(5) - (10π)/(5) = -(4π)/(5). Since secant is an even function, sec(-(4π)/(5)) is equivalent to sec((4π)/(5)), which is not equivalent to sec((π)/(5)).Evaluate Fourth Option: Evaluate the fourth option, sec((4π)/(5)). This is not equivalent to sec((π)/(5)) because (4π)/(5) is not equal to (π)/(5) plus any multiple of 2π.Compare Results: Compare the results from Steps 2, 3, 4, and 5. The only equivalent expressions to sec((π)/(5)) are sec(-(π)/(5)) from Step 3 and sec((14π)/(5)) from Step 2, which simplifies to sec((4π)/(5)). However, since sec((4π)/(5)) is not equivalent to sec((π)/(5)), there must be a mistake in the previous steps.

Which of the following is equivalent to $\sec \frac{\pi}{5}$ ? $\newline$ $\sec \frac{14 \pi}{5}$ $\newline$ $\sec \left(-\frac{\pi}{5}\right)$ $\newline$ $\sec \frac{6 \pi}{5}$ $\newline$ $\sec \frac{4 \pi}{5}$

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