The Sum of Tangent and Cotangent Is at Least 2

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Let be an angle between 0 and . In both images, the length of the red segment is and the length of the green segment is . In the left image, draw the radial line forming angle with the positive axis to intersect the unit circle at the black point. Then the segment of the tangent to the unit circle at the black point between the coordinate axes has length . When a circle is drawn with this diameter, it is clear that the radius of this circle is larger than that of the unit circle, thus illustrating the inequality .

Contributed by: Chris Boucher (February 2019)
After work by: Ángel Plaza
Open content licensed under CC BY-NC-SA


Details

Reference

[1] A. Plaza, "Proof Without Words: Tangent Plus Cotangent Is Greater or Equal Than 2," Mathematics Magazine, 91(5), 2018 p. 363. doi:10.1080/0025570X .2018.1465772.


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