Which of the following is not equivalent to cot ((3pi)/(8))? cot(-(13 pi)/(8)) cot ((5pi)/(8)) cot ((19 pi)/(8)) cot ((11 pi)/(8))

Question

Which of the following is not equivalent to  cot ((3pi)/(8))?  cot(-(13 pi)/(8))  cot ((5pi)/(8))  cot ((19 pi)/(8))  cot ((11 pi)/(8))

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Periodicity of Cotangent Function: Understand the cotangent function's periodicity. The cotangent function has a period of π, which means cot(θ) = cot(θ + kπ) for any integer k.Evaluate -(13pi)/(8): Evaluate cot(-(13pi)/(8)). Using the periodicity of the cotangent function, we can add π to the angle until it is positive: cot(-(13pi)/(8)) = cot(-(13pi)/(8) + 2π) = cot((3pi)/(8)) This is because adding 2π is the same as adding π twice, which does not change the value of the cotangent function.Evaluate (5pi)/(8): Evaluate cot((5pi)/(8)). Using the periodicity of the cotangent function, we can add π to the angle: cot((5pi)/(8)) = cot((5pi)/(8) + π) = cot((13pi)/(8)) This is not equivalent to cot((3pi)/(8)) because we have added π to a different starting angle.Evaluate (19pi)/(8): Evaluate cot((19pi)/(8)). Using the periodicity of the cotangent function, we can subtract π from the angle: cot((19pi)/(8)) = cot((19pi)/(8) - 2π) = cot((3pi)/(8)) This is because subtracting 2π is the same as subtracting π twice, which does not change the value of the cotangent function.Evaluate (11pi)/(8): Evaluate cot((11pi)/(8)). Using the periodicity of the cotangent function, we can subtract π from the angle: cot((11pi)/(8)) = cot((11pi)/(8) - π) = cot((3pi)/(8)) This is because subtracting π from the angle does not change the value of the cotangent function.Determine Non-Equivalent Expression: Determine which expression is not equivalent. From the previous steps, we have determined that cot(-(13pi)/(8)), cot((19pi)/(8)), and cot((11pi)/(8)) are all equivalent to cot((3pi)/(8)). However, cot((5pi)/(8)) is not equivalent because it is the cotangent of a different angle that is not simply a multiple of π away from (3pi)/(8).

Which of the following is not equivalent to $\cot \frac{3 \pi}{8} ?$ $\newline$ $\cot \left(-\frac{13 \pi}{8}\right)$ $\newline$ $\cot \frac{5 \pi}{8}$ $\newline$ $\cot \frac{19 \pi}{8}$ $\newline$ $\cot \frac{11 \pi}{8}$

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