Which of the following is equivalent to cot ((pi)/(4)) ? cot(-(pi)/(4)) cot ((7pi)/(4)) cot ((3pi)/(4)) cot(-(7pi)/(4))

Understand cotangent function: Understand the cotangent function and its periodicity. The cotangent function, cot(x), is the reciprocal of the tangent function, tan(x). It has a period of π, which means cot(x + π) = cot(x). We will use this property to find the equivalent expression.Evaluate cot(pi/4): Evaluate cot((pi)/(4)). We know that tan((pi)/(4)) = 1 because the tangent of 45 degrees (or π/4 radians) is 1. Therefore, cot((pi)/(4)) = 1/1 = 1.Compare cot(pi/4) with cot(-pi/4): Compare cot((pi)/(4)) with cot(-(pi)/(4)). Using the cotangent's odd symmetry, cot(-x) = -cot(x), we find that cot(-(pi)/(4)) = -cot((pi)/(4)) = -1. This is not equivalent to cot((pi)/(4)).Compare cot(pi/4) with cot(7pi/4): Compare cot((pi)/(4)) with cot((7pi)/(4)). Since cot(x) has a period of π, adding π to (pi)/(4) gives us cot((pi)/(4) + π) = cot((5pi)/(4)). Adding another π gives us cot((5pi)/(4) + π) = cot((7pi)/(4)). Therefore, cot((7pi)/(4)) = cot((pi)/(4)) = 1.Compare cot(pi/4) with cot(3pi/4): Compare cot((pi)/(4)) with cot((3pi)/(4)). The cotangent of (3pi)/(4) is the reciprocal of the tangent of (3pi)/(4). Since tan((3pi)/(4)) = -1, cot((3pi)/(4)) = -1. This is not equivalent to cot((pi)/(4)).Compare cot(pi/4) with cot(-7pi/4): Compare cot((pi)/(4)) with cot(-(7pi)/(4)). Using the cotangent's odd symmetry again, cot(-x) = -cot(x), we find that cot(-(7pi)/(4)) = -cot((7pi)/(4)). Since we established that cot((7pi)/(4)) = 1, cot(-(7pi)/(4)) = -1. This is not equivalent to cot((pi)/(4)).

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Algebra 1

Multiplication with rational exponents

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Q. Which of the following is equivalent to

\cot \frac{\pi}{4}

\newline

\cot \left(-\frac{\pi}{4}\right)

\newline

\cot \frac{7 \pi}{4}

\newline

\cot \frac{3 \pi}{4}

\newline

\cot \left(-\frac{7 \pi}{4}\right)

Understand cotangent function: Understand the cotangent function and its periodicity. $\newline$ The cotangent function, $\cot(x)$ , is the reciprocal of the tangent function, $\tan(x)$ . It has a period of $\pi$ , which means $\cot(x + \pi) = \cot(x)$ . We will use this property to find the equivalent expression.
Evaluate $\cot(\pi/4)$ : Evaluate $\cot(\frac{\pi}{4})$ . We know that $\tan(\frac{\pi}{4}) = 1$ because the tangent of $45$ degrees (or $\pi/4$ radians) is $1$ . Therefore, $\cot(\frac{\pi}{4}) = \frac{1}{1} = 1$ .
Compare $\cot(\frac{\pi}{4})$ with $\cot(-\frac{\pi}{4})$ : Compare $\cot\left(\frac{\pi}{4}\right)$ with $\cot\left(-\frac{\pi}{4}\right)$ . Using the cotangent's odd symmetry, $\cot(-x) = -\cot(x)$ , we find that $\cot\left(-\frac{\pi}{4}\right) = -\cot\left(\frac{\pi}{4}\right) = -1$ . This is not equivalent to $\cot\left(\frac{\pi}{4}\right)$ .
Compare $\cot(\pi/4)$ with $\cot(7\pi/4)$ : Compare $\cot(\left(\pi\right)/(4))$ with $\cot(\left(7\pi\right)/(4))$ . $\newline$ Since $\cot(x)$ has a period of $\pi$ , adding $\pi$ to $(\pi)/(4)$ gives us $\cot((\pi)/(4) + \pi) = \cot((5\pi)/(4))$ . Adding another $\pi$ gives us $\cot(7\pi/4)$ $0$ . Therefore, $\cot(($7$\pi)/($4$)) = \cot((\pi)/($4$)) = $1$.
Compare $\cot(\frac{\pi}{4})$ with $\cot(\frac{3\pi}{4})$: Compare $\cot(\left(\frac{\pi}{4}\right))$ with $\cot(\left(\frac{3\pi}{4}\right))$. The cotangent of $\left(\frac{3\pi}{4}\right)$ is the reciprocal of the tangent of $\left(\frac{3\pi}{4}\right)$. Since $\tan(\left(\frac{3\pi}{4}\right)) = -1$, $\cot(\left(\frac{3\pi}{4}\right)) = -1$. This is not equivalent to $\cot(\left(\frac{\pi}{4}\right))$.
Compare $\cot(\pi/4)$ with $\cot(-7\pi/4)$: Compare $\cot(\left(\pi\right)/(4))$ with $\cot\left(-(7\pi)/(4)\right)$. Using the cotangent's odd symmetry again, $\cot(-x) = -\cot(x)$, we find that $\cot\left(-(7\pi)/(4)\right) = -\cot\left((7\pi)/(4)\right)$. Since we established that $\cot\left((7\pi)/(4)\right) = 1$, $\cot\left(-(7\pi)/(4)\right) = -1$. This is not equivalent to $\cot\left((\pi)/(4)\right)$.