Questions: Use the properties of inverse functions f(f^(-1)(x))=x for all x in the domain of f^(-1) and f^(-1)(f(x))=x for all x in the domain of f, as well as the definitions of the inverse cotangent function, to find the exact value of the expression. cot(cot^(-1) 25 pi) cot(cot^(-1) 25 pi) = (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.)

$Use the properties of inverse functions f(f^(-1)(x))=x for all x in the domain of f^(-1) and f^(-1)(f(x))=x for all x in the domain of f, as well as the definitions of the inverse cotangent function, to find the exact value of the expression. cot(cot^(-1) 25 pi) cot(cot^(-1) 25 pi) = (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.)$

Transcript text: 4 Trigonometry Justin McLeod 10/17/24 10:15 PM omework: Section 5.7 Question 28, 5.7.55 HW Score: 80.56\%, 29 of 36 points Points: 0 of 1 Save tion list estion 25 stion 26 stion 27 stion 28 Use the properties of inverse functions $f\left(f^{-1}(x)\right)=x$ for all $x$ in the domain of $f^{-1}$ and $f^{-1}(f(x))=x$ for all $x$ in the domain of f , as well as the definitions of the inverse cotangent function, to find the exact value of the expression. \[ \cot \left(\cot ^{-1} 25 \pi\right) \] $\cot \left(\cot ^{-1} 25 \pi\right)=$ $\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

Solution

Solution Steps

To solve the expression $\cot \left(\cot^{-1} 25 \pi\right)$, we can use the property of inverse functions. Specifically, the property states that for a function $f$ and its inverse $f^{-1}$, $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$. Here, $\cot$ and $\cot^{-1}$ are inverse functions, so $\cot(\cot^{-1}(x)) = x$ for all $x$ in the domain of $\cot^{-1}$. Therefore, the expression simplifies directly to $25\pi$.

Step 1: Understanding the Expression

We need to evaluate the expression $ \cot \left( \cot^{-1} 25 \pi \right) $. By the properties of inverse functions, we know that $ \cot(\cot^{-1}(x)) = x $ for all $ x $ in the domain of $ \cot^{-1} $.

Step 2: Applying the Inverse Function Property

Using the property mentioned, we can simplify the expression: \[ \cot \left( \cot^{-1} 25 \pi \right) = 25 \pi \]

Step 3: Final Result

The exact value of the expression is $ 25 \pi $.