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We have to verify sin x < 1 / 2 cos x.
sin x < 1 / 2 cos x
=> 2 sin x * cos x < 1
=> sin 2x < 1
But sin 2x can also be equal to 1.
Note: Therefore the inequality should be sin x < = 1/2 cos x.
We therefore prove that sin x < = 1/ 2 cos x.
We'll multiply by 2 cos x both sides:
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.
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