Which of the following inequalities have no solution over all real numbers? Check all that apply.

The cosine and sine functions have values only from −1 to 1. So cos(x) < −2 is impossible, and cos(x) − 1 > 0.5 would require cos(x) > 1.5, also impossible. For D, csc(x) = 1/sin(x) can only be at most −1 or at least 1, so it cannot lie strictly between −1 and 1. The other inequalities do have solutions: sine can exceed 2/3, secant can be less than 1, and tangent can be less than −4.

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