Using a calculator, determine $\cot\frac{\pi}{64}$. Round the result to four decimal places.
Adele followed these steps:
(1) She converted the angle from radians to degrees: $$\cot\frac{\pi}{64}=\cot\left(\frac{45}{16}\right)^\circ$$
(2) She stated that for any angle $x$ where the functions $\tan(x)$ and $\cot(x)$ exist, these functions are reciprocals of each other.
(3) Based on statement (2), she concluded that: $$\cot\left(\frac{45}{16}\right)^\circ=\tan\left(\frac{16}{45}\right)^\circ$$
(4) Therefore, using a calculator, she calculated: $$\tan\left(\frac{16}{45}\right)^\circ\approx 0.0062$$ (rounded to four decimal places) and declared this as the final result.
However, her result is incorrect. In which step did Adele make a mistake?
The mistake is in step (1). The conversion of the angle to degrees was unnecessary, but since she chose to convert it, it should have been done correctly.
The mistake is in step (2). The claim that $\tan(x)$ and $\cot(x)$ are always reciprocal functions is not true.
The mistake is in step (3). The equation $\cot\left(\frac{45}{16}\right)^\circ=\tan\left(\frac{16}{45}\right)^\circ$ is not true.
The mistake is in step (4). Adele made a mistake when using the calculator. She likely had the calculator set to radians instead of degrees when entering the angle in degrees.
For every angle $x$ for which the functions $\tan(x)$ and $\cot(x)$ are defined, these functions are reciprocals of each other, meaning: $$\cot x=\frac{1}{\tan x}$$ Therefore $$\cot\frac{\pi}{64}=\frac{1}{\tan\frac{\pi}{64}}\approx20.3555,$$ or using the degree conversion correctly: $$\cot\frac{\pi}{64}=\cot\left(\frac{45}{16}\right)^\circ=\frac{1}{\tan\left(\frac{45}{16}\right)^\circ }\approx20.3555.$$