Given equations:
- acos(theta) - bcos(theta) = x
- asin(theta) + bcos(theta) = y

To find x and y:
We are given two equations involving trigonometric functions and we need to solve for x and y.

Step 1: Express cos(theta) in terms of sin(theta)
We can use the trigonometric identity sin^2(theta) + cos^2(theta) = 1 to express cos(theta) in terms of sin(theta):
cos^2(theta) = 1 - sin^2(theta)
cos(theta) = sqrt(1 - sin^2(theta))

Step 2: Substitute cos(theta) in the given equations
Substituting cos(theta) = sqrt(1 - sin^2(theta)) in the first equation:
acos(theta) - bcos(theta) = x
a(sqrt(1 - sin^2(theta))) - b(sqrt(1 - sin^2(theta))) = x
(sqrt(1 - sin^2(theta)))(a - b) = x

Substituting cos(theta) = sqrt(1 - sin^2(theta)) in the second equation:
asin(theta) + bcos(theta) = y
asin(theta) + b(sqrt(1 - sin^2(theta))) = y

Step 3: Solve for sin(theta)
Let's solve the second equation for sin(theta):
asin(theta) + b(sqrt(1 - sin^2(theta))) = y
asin(theta) = y - b(sqrt(1 - sin^2(theta)))
sin(theta) = (y - b(sqrt(1 - sin^2(theta))))/a

Step 4: Substitute sin(theta) in the first equation
Substituting sin(theta) = (y - b(sqrt(1 - sin^2(theta))))/a in the first equation:
(sqrt(1 - ((y - b(sqrt(1 - sin^2(theta))))/a)^2))(a - b) = x

Step 5: Simplify the equation
To simplify the equation, let's substitute a^2 and b^2 with x^2 and y^2 respectively:
(sqrt(1 - ((y - b(sqrt(1 - ((y - b(sqrt(1 - sin^2(theta))))/a)^2))))^2))(a - b) = x
(sqrt(1 - ((y - b(sqrt(1 - ((y - b(x^2))^2))))^2))(a - b) = x

Hence, we have the simplified equation a^2 b^2 = x^2 y^2.

Conclusion:
By following the given steps, we derived the equation a^2 b^2 = x^2 y^2. This equation represents the relationship between the given variables a, b, x, and y.