prove the inequality sin x + tan x > 2x for x �¸ (0, pi/2)

Question

i don't know what i must do with this equation, can someone explain this to me step by step? >.<

Answer 1

let's look at

y = sinx + tanx - 2x

by graphing it on Wolfram http://www.wolframalpha.com/input/?i=sin%28x%29+%2B+tan%28x%29+-+2x

shows a solution at x = 0 and the next one at x = ±4.615
your domain is between 0 and π/2
so x=0 will work for sinx + tanx - 2x = 0

so for > 0 we would have

0 < x ≤ π/2

PS, to actually solve this would be a real mess.
you could use something like Newton's Method but I think the iteration is very slow