If we'll divide by cos x, we'll get:
2 sin x/cos x < 1
2 tan x < 1
tan x < 1/2
Since the tangent function is not limited, this inequality is not holding for any value of x.
I'll suggest to analyze the following inequality instead:
2sin x< 1/cos x
2 sin x * cos x<1
Instead of the value 1, we'll put the fundamental relation of trigonometry:
1= (sin x)^2 + (cos x)^2
The inequality will become:
2 sin x * cos x < (sin x)^2 + (cos x)^2
We'll subtract 2sin x*cos x both sides:
(sin x)^2 -2 sin x * cos x + (cos x)^2>0
The expression from the left side is a perfect square:
(sin x - cos x)^2 > 0, true, for any real value of x.