<s>[INST] Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian? Testing with my calculator in degree mode, I have found the following to be true:
$$\tan \left(90 - \frac{1}{10^n}\right) \approx \frac{180}{\pi} \times 10^n, n \in \mathbb{N}$$
Why is this? What is the proof or explanation?
 [/INST] The power series for cotangent (for $x$ in radians) is
$$\cot x = \frac{1}{x} - \frac{1}{3}x - \frac{1}{45}x^3 - \frac{2}{945}x^5 - \cdots \approx \frac{1}{x} \; \text{for $x$ small}$$
So,
$$\tan\left( 90^\circ - \frac{1^\circ}{10^n} \right) 
= \cot\frac{1^\circ}{10^n} 
= \cot\frac{\pi/180}{10^n} 
\approx \frac{10^n}{\pi/180} = \frac{180}{\pi}\times 10^n$$
Congratulations for noticing the pattern on your calculator!
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