Final answer:
To find cos(x + y), use the given values of cos x and tan y, account for the specific quadrants to determine the signs of sin x and cos y, then apply the cosine sum formula.
Step-by-step explanation:
To calculate cos(x + y) given that cos x = 15/17 and tan y = 4/3, we can use trigonometric identities and the information about the angles' quadrants to find cos y and sin x. Since 3π/2 < x < 2π, x is in the fourth quadrant where cosine is positive and sine is negative. As π < y < 3π/2, y is in the third quadrant where tangent is positive, but sine and cosine are negative.
We use the Pythagorean identity to find sin x and cos y: sin^2 x = 1 - cos^2 x and cos^2 y = 1/(1 + tan^2 y). Now, sin x = √(1 - (15/17)^2) and it will be negative because of the quadrant x is in. For cos y, cos y = -√(1/(1 + (4/3)^2)) because cos y is negative in the third quadrant.
Lastly, we use the cosine sum formula: cos(x + y) = cos x × cos y - sin x × sin y, and substitute the values we found to calculate the answer.