Question

If x = 3 sin θ + 4 cos θ and y = 3 cos θ – 4 sin θ then prove that x² + y² = 25.

Answer 1

Given, x = 3 sin θ + 4 cos θ and y = 3 cos θ – 4 sin θ

∴ x² + y² = (3 sin θ + 4 cos θ)² + (3 cos θ – 4 sin θ)²

= (9 sin²θ + 16 cos²θ + 24 sinθ cosθ) + (9 cos²θ + 16 sin²θ – 24 sinθ cosθ)

= 9(sin²θ + cos²θ) + 16(sin²θ + cos²θ)

= 9 + 16   ...[∵ sin²θ + cos²θ = 1]

= 25

Hence Proved.