Which of the following trigonometric inequalities has no solution over the interval 0 ≤ x ≤ 2pi radians?

A. cos (x) <1

B. cos (x/2) <1

C. csc (x) <1

D. csc (x/2) <1

A. cos (x) <1

Step-by-step explanation:

According to the first inequality

cos (x) <1

x < arccos 1

x<0

This therefore does not have a solution within the range 0 ≤ x ≤ 2pi

x cannot be leas than 0. According to the range not value, 0≤x which is equivalent to x≥0. Thus means otvis either x = 0 or x> 0.

For the second option

. cos (x/2) <1

x/2< arccos1

x/2<0

x<0

This inequality also has solution within the range 0 ≤ x ≤ 2pi since 0 falls within the range of values.

For the inequality csc (x) <1

1/sin (x) < 1

1< sin (x)

sinx>1

x>arcsin1

x>90°

x>π/2

This inequality also has solution within the range 0 ≤ x ≤ 2pi since π/2 falls within the range of values

For the inequality csc (x/2) <1

1/sin (x/2) < 1

1< sin (x/2)

sin (x/2) > 1

x/2 > arcsin1

X/2 > 90°

x>180°

x>π

This value of x also has a solution within the range.

Therefore option A is the only inequality that does not have a solution with the range.