The Sum of Tangent and Cotangent Is at Least 2

α

1.1333

Let

α

be an angle between 0 and

π/2

. In both images, the length of the red segment is

tanα

and the length of the green segment is

cotα

. In the left image, draw the radial line forming angle

α

with the positive

x

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

tan(α)+cot(α)

. 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

tan(α)+cot(α)≥2

.

References

[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.

Permanent Citation

Chris Boucher, Ángel Plaza

"The Sum of Tangent and Cotangent Is at Least 2"
http://demonstrations.wolfram.com/TheSumOfTangentAndCotangentIsAtLeast2/
Wolfram Demonstrations Project
Published: February 21, 2019