Final answer:
Given sec(θ) = 5/3 in the fourth quadrant, cos(θ) = 3/5, so option A is correct. Options B, C, and D are incorrect as the correct value for sin(θ) is -4/5, not -2/5.
Step-by-step explanation:
When given sec(θ)= 5/3 and knowing that θ is in the fourth quadrant, we can determine various trigonometric functions of θ. In the fourth quadrant, cosine is positive and sine is negative. The secant function is the reciprocal of the cosine function, so cos(θ) = 1/sec(θ) = 3/5, which makes option A correct. To find sin(θ), we use the Pythagorean identity sin2(θ) + cos2(θ) = 1. If cos(θ) = 3/5, then sin(θ)= -√(1 - (3/5)2) = -√(1 - 9/25) = -√(16/25) = -4/5. Therefore, sin(θ) = -(4/5), not -(2/5), so option D is incorrect. Since csc(θ)= 1/sin(θ) and tan(θ) = sin(θ)/cos(θ), neither B nor C are correct due to our calculated sin(θ).