Question

If x = a sin θ + b cos θ, y = a cos θ – b sin θ, prove that x² + y² = a² + b².

Answer 1

Given: x = a sin θ + b cos θ, y = a cos θ – b sin θ

To Prove: x² + y² = a² + b²

Proof [Step-wise]:

1. Compute: 

x² + y² = (a sin θ + b cos θ)² + (a cos θ – b sin θ)²

2. Expand both squares:

= a² sin²θ + 2ab sin θ cos θ + b² cos²θ

a² cos²θ – 2ab sin θ cos θ + b² sin²θ

3. Cancel the cross terms (+2ab sin θ cos θ and –2ab sin θ cos θ):

= a²(sin²θ + cos²θ) + b²(cos²θ + sin²θ)

4. Use the Pythagorean identity sin²θ + cos²θ = 1 to get:

= a² × 1 + b² × 1

= a² + b²

Therefore, x² + y² = a² + b², as required.