prove that (tanx)/x is greater than x/(sinx)?

Answer 1

It isn't necessarily.
For example,
tan(2)/2 is negative. (-1.0925...)
2/sin(2) is positive. (2.1995...)

Note:jdsheth2004 has used degrees rather than radians in the trig. calculations; nevertheless the hypothesis still remains false.

Answer 2

It is true over the interval -pi/2 < x
However consider x = 2pi/3

Here tan 2pi/3 = -square root 3 and therefore [tan (2pi/3)]/(2pi/3) is negative (less than one).

But sin 2pi/3 = (square root 3)/2, so
(2pi/3)/sin (2pi/3) is positive.

Therefore in the interval [pi/2

Answer 3

(tanx)/x
=(sinx/cosx)/x
=sinx/cosx*x
consider, x=1,then
(tanx)/x
=0.0175/1
=0.0175
x/(sinx)
=1/sin 1
=57.29
So, here it is opposite result than what you have asked.