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.)

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 $25\pi$.

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 $\cot^{-1}$.

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

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

Final Answer