Question

If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.

Answer 1

Given 3 sin θ + 5 cos θ = 5

Squaring on both sides for both the equations 

⇒ 9 sin²θ + 25 cos²θ + 30 sinθ cosθ = 25

⇒ 25 sin²θ + 9 cos²θ − 30 sinθ cosθ = x²

Adding the equations;

⇒ 34 (sin²θ + cos²θ) = 25 + x²

⇒ x² = 34 − 25 = 9

⇒ x = ±3

∴ 5 sinθ − 3 cosθ = ±3

Hence proved.