Question

Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.

Answer 1

Given: sin θ + 2 cos θ = 1

Squaring on both sides,

(sin θ + 2 cos θ)² = 1

⇒ sin² θ + 4 cos² θ + 4sin θ cos θ = 1

Since, sin² θ = 1 – cos² θ and cos² θ = 1 – sin² θ

⇒ (1 – cos² θ) + 4(1 – sin² θ) + 4sin θ cos θ = 1

⇒ 1 – cos² θ + 4 – 4 sin² θ + 4sin θ cos θ = 1

⇒ – 4 sin² θ – cos² θ + 4sin θ cos θ = – 4

⇒ 4 sin² θ + cos² θ – 4sin θ cos θ = 4

We know that,

a² + b² – 2ab = (a – b)²

So, we get,

(2sin θ – cos θ)² = 4

⇒ 2sin θ – cos θ = 2

Hence proved.