This is a problem number 7 from ISI B.Math 2007 based on an inequality related to (sin x)/x function. Try out this problem.
Problem: An inequality related to (sin x)/x function
Let . Prove that .
We consider the function . The first derivative of this function is In the interval the numerator is always negative as x is less than tan x.
Hence f(x) is a monotonically decreasing function in the given interval. Hence f(x) attains least value at which equals
Therefore in the given interval.