Question

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

Answer 1

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

To Prove: a² + b² = x² + y²

Proof [Step-wise]:

1. Square the first equation:

(x cos θ + y sin θ)² = a²

⇒ x² cos² θ + 2xy sin θ cos θ + y² sin² θ = a²

2. Square the second equation:

(x sin θ – y cos θ)² = b²

⇒ x² sin² θ – 2xy sin θ cos θ + y² cos² θ = b²

3. Add the two results:

[x² cos² θ + x² sin² θ] + [y² sin² θ + y² cos² θ]

[2xy sin θ cos θ – 2xy sin θ cos θ] = a² + b²

4. Use sin² θ + cos² θ = 1 and cancellation of the cross terms:

x²(1) + y²(1) = a² + b²

⇒ x² + y² = a² + b²

Hence a² + b² = x² + y².