Compare the numbers 3tan 2 and 2tan 3

We'll choose a function whose domain of definition is (0 ;
3pi/180).

The function is f(x) = (tan
x)/x

We'll do the first derivative test to verify the if
the function is monotonic. We'll apply the quotient
rule:

f'(x) = [(tan x)'*x - (tan
x)*(x)']/x^2

f'(x) = [x/(cos x)^2 - sin x/cos
x]/x^2

f'(x) = (x - sin x*cos x)/x^2*(cos
x)^2

We notice that the numerator is positive for any value
from the domain of definition and the denominator is always
positive.

Then f'(x)>0 => the function is
monotonically increasing.

If so, for  x=2pi/180 <
x=3pi/180 => f(2pi/180) < f(3pi/180)

But
f(2pi/180) = tan (2pi/180)/(2pi/180)

f(3pi/180) = tan
(3pi/180)/(3pi/180)

tan (2pi/180)/(2pi/180) < tan
(3pi/180)/(3pi/180)

We'll simplify by
pi/180:

(tan 2)/2 < (tan
3)/3

We'll cross multiply and we'll
get:

3*tan 2 < 2*tan
3

Comparing the given values, we have found
the inequality: 3*tan 2 < 2*tan 3.