How to prove that (tan b / tab a) > (b/a) whenever 0 < a < b < π/2?

Question

I am able to show that tan x > x for 0 < x < π/2 but this doesn't seems to get me through to the final conclusion I'm looking for.

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Answer 1 · 2017-08-07T23:42:46

Consider the function:

#f(x) = tanx/x # in the interval #(0,pi/2)#

we have that:

#(df)/dx = d/dx (tanx/x) = (x/cos^2x-tanx)/x^2 = (x-sinxcosx)/(x^2cos^2x)#

Now, as #x > sinx # and #cosx < 1#, then we have that:

#x - sinx cosx > 0#

and

#d/dx (tanx/x) > 0#

so that the function is strictly increasing and:

#b > a => tanb/b > tana/a#

As for #0 < a < b < pi/2# we have that #b > 0# and #tana > 0# this is equivalent to:

#tanb/tana > b/a#