One small step

Weird fact of the day: arctan(x)+arctan(x^{-1})=sgn(x)\frac{\pi}{2}, x \neq 0.

Why is it weird? Because the sum of two clearly non-constant functions is constant. Well, more or less so.

How to prove it? Differentiate.

How can I use it? You can confuse people with it: “Hello, clerk, I want 2arctan(e^{\pi})/\pi+2arctan(e^{-\pi})/\pi” tickets to this movie.” Also, it works as a very fancy, if slightly incorrect way to write signum function.

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2 Responses to One small step

  1. christopherolah says:

    Hm… We can make a geometric proof of this as well. Consider the interpretation of tan(x) as the length of the line running tangent to the unit circle at θ to the horizontal axis and tan(θ)⁻¹=cot(θ) as same line but to the vertical axis.

    Then atan(x) is the angle, θ, such that the first line segment has a length of l. atan(x⁻¹) is the angle so that the second line has a length of l, which is clearly π/2 – θ. So their sum is π/2. An identical argument holds if l<0, except the sign flips.

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