Question

If sin θ + 2 cos θ = 1 prove that 2 sin θ – cos θ = 2.

Answer 1

Given: sin θ + 2 cos θ = 1

To Prove: The statement 2 sin θ – cos θ = 2 as stated is not true for every θ satisfying the given relation; instead (2 sin θ – cos θ) = ±2. (We will show (2 sin θ – cos θ)² = 4 and obtain the two possible values ±2; the value +2 occurs only for a specific θ.)

Proof [Step-wise]:

1. Put s = sin θ and c = cos θ.

The given is s + 2c = 1.

2. Square both sides:

(s + 2c)² = 1² 

⇒ s² + 4c² + 4sc = 1

3. Replace s² by 1 – c²: 

(1 – c²) + 4c² + 4sc = 1 

⇒ 3c² + 4sc = 0

⇒ c(3c + 4s) = 0

4. So either (A) c = 0 or (B) 3c + 4s = 0.

5. Case A: c = 0.

Then from s + 2c = 1 we get s = 1. 

Hence 2s – c = 2·1 – 0 = 2.

6. Case B: 3c + 4s = 0.

Solve this together with s + 2c = 1.

From 3c + 4s = 0 we get s = `(-3c)/4`.

Substitute into s + 2c = 1: `(-3c)/4 + 2c = 1`

⇒ `(5c)/4 = 1`

⇒ `c = 4/5` and then `s = -3/5` 

Compute `2s - c = 2(-3/5) - 4/5`

= `-6/5 - 4/5`

= `-10/5`

= –2

7. From cases A and B we have shown for any θ satisfying the given that 2s – c is either 2 or –2. 

Equivalently, (2 sin θ – cos θ)² = 4, so 2 sin θ – cos θ = ±2.