Verify an inequality2sinx<cosx

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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