If x = a cos θ + b sin θ and y = b cos θ − a sin θ, what is the simplified value of x^2 + y^2 expressed purely in terms of a and b?

Introduction / Context:
This question checks your understanding of trigonometric expressions combined with algebra and the Pythagorean identity. It is a classic example where trigonometric terms vanish after simplification, leaving a result that depends only on algebraic parameters a and b. Such forms appear in rotations and transformations in coordinate geometry and physics.

Given Data / Assumptions:

• x = a cos θ + b sin θ.
• y = b cos θ − a sin θ.
• We must compute x^2 + y^2.
• Angle θ is real and sin θ and cos θ are defined.

Concept / Approach:
Key ideas:

• Expand x^2 and y^2 using algebraic identities.
• Combine like terms in cos^2 θ, sin^2 θ, and sin θ cos θ.
• Use the identity sin^2 θ + cos^2 θ = 1.
• Observe that cross terms cancel out, leaving an expression in a and b only.

Step-by-Step Solution:
Compute x^2: x^2 = (a cos θ + b sin θ)^2 = a^2 cos^2 θ + 2ab sin θ cos θ + b^2 sin^2 θ. Compute y^2: y^2 = (b cos θ − a sin θ)^2 = b^2 cos^2 θ − 2ab sin θ cos θ + a^2 sin^2 θ. Add x^2 and y^2. x^2 + y^2 = a^2 cos^2 θ + 2ab sin θ cos θ + b^2 sin^2 θ + b^2 cos^2 θ − 2ab sin θ cos θ + a^2 sin^2 θ. Notice that the +2ab sin θ cos θ and −2ab sin θ cos θ terms cancel. Group a terms: a^2 cos^2 θ + a^2 sin^2 θ = a^2 (cos^2 θ + sin^2 θ) = a^2. Group b terms: b^2 sin^2 θ + b^2 cos^2 θ = b^2 (sin^2 θ + cos^2 θ) = b^2. Therefore x^2 + y^2 = a^2 + b^2.

Verification / Alternative check:
Pick simple values, for example a = 1, b = 2, and θ = 30°. Compute x and y numerically and then find x^2 + y^2. Separately compute a^2 + b^2 = 1^2 + 2^2 = 5. Both calculations will give the same numerical result, confirming the formula.

Why Other Options Are Wrong:
Option a: ab would imply all squared terms vanish, which is not supported by the algebra.
Option c: a^2 − b^2 would require different cancellation and sign patterns, which do not occur here.
Option d: 1 would suggest the expression is independent of a and b, which is impossible because x and y scale with a and b.
Option e: (a + b)^2 = a^2 + 2ab + b^2, but the cross term 2ab does not remain in the simplified expression.

Common Pitfalls:
A common mistake is mishandling the cross terms 2ab sin θ cos θ and failing to notice that they cancel when x^2 and y^2 are added. Another frequent error is not applying sin^2 θ + cos^2 θ = 1 correctly, which leaves unnecessary trigonometric expressions in the final result.

Final Answer:
The value of x^2 + y^2 is a^2 + b^2.