Which of the following inequalities is satisfied for every real value of x ?
A) 2sin²x − sin x − 1 > 0
B) sin²x + sin x − 2 ≤ 0
C) 2cos²x + cos x − 1 < 0
D) tan²x − 2tan x + 1 > 0
E) cos²x − cos x − 2 ≥ 0
IMAT 2025 · Question 49
Q49 - Trigonometric equations and inequalities
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The correct answer is B) sin²x + sin x − 2 ≤ 0.
Factorise each one with sin x, cos x or tan x as the variable.
For B, s² + s − 2 = (s + 2)(s − 1). Since sin x is between −1 and 1, (s + 2) is always positive and (s − 1) is never positive, so the product is never above zero. It holds for every x.
The others all fail somewhere. A fails at sin x = 0, C at cos x = 1, D at tan x = 1, and E only holds when cos x = −1.
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