Trigonometry and Inequality

Geometry Level 4

Let ρ0,1\rho \neq 0, 1 be a real number. How many ϕ[0,2π)\phi \in [0, 2\pi) exists such that the statement ρcosθ=ρsin(θ+ϕ)\rho \cos\theta = \sqrt{\rho}\sin(\theta + \phi) holds for all real θ\theta?

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