Answer:√(4/5)Step-by-step explanation:First, let's use reflection property to find tan θ.tan(-θ) = 1/2-tan θ = 1/2tan θ = -1/2Since tan θ < 0 and sec θ > 0, θ must be in the fourth quadrant.Now let's look at the problem we need to solve:sin(5π/2 + θ)Use angle sum formula:sin(5π/2) cos θ + sin θ cos(5π/2)Sine and cosine have periods of 2π, so:sin(π/2) cos θ + sin θ cos(π/2)Evaluate:(1) cos θ + sin θ (0)cos θWe need to write this in terms of tan θ. We can use Pythagorean identity:1 + tan² θ = sec² θ1 + tan² θ = (1 / cos θ)²±√(1 + tan² θ) = 1 / cos θcos θ = ±1 / √(1 + tan² θ)Plugging in:cos θ = ±1 / √(1 + (-1/2)²)cos θ = ±1 / √(1 + 1/4)cos θ = ±1 / √(5/4)cos θ = ±√(4/5)Since θ is in the fourth quadrant, cos θ > 0. So:cos θ = √(4/5)Or, written in proper form:cos θ = (2√5) / 5