Cos>2

For real x, there are no solutions.

Reason: cos x ∈ [-1, 1] for all real x, so cos x can never exceed 2.

If you allow complex x, there are solutions. For example, along the imaginary axis:

• x = i y with y > arcosh(2) (where arcosh is the inverse hyperbolic cosine),
• cos(i y) = cosh(y) > 2.

More generally, cos(2πk ± i y) = cosh(y), so any z = 2πk ± i y with k ∈ Z and y > arcosh(2) satisfies cos z > 2.