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209 geometry/trigonometry/inequality.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

209 geometry/trigonometry/inequality.p


Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4.

geometry/trigonometry/inequality.s

The function f(x) = x^(1/sqrt(1-x^2)) is monotonically increasing for
0 < x < 1, easily verified by taking the derivative.
Since 0 < sin x < cos x < 1 for 0 < x < pi/4, f(sin x) < f(cos x).
But f(sin x) = (sin x)^(1/cos x) and f(cos x) = (cos x)^(1/sin x).
Raising both sides to the power (cos x.sin x), we get the desired
result.

 

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