Question

If sin A + 2cos A = 1

Prove That 2sin A - cos A = 2

Please explain with full steps

Answer 1

Given, sinθ+2cosθ=1

On squaring both sides, we get;

(sinθ+2cosθ)2=1

⇒sin2θ+4cos2θ+4sinθ.cosθ=1

⇒(1−cos2θ)+4(1−sin2θ)+4sinθ.cosθ=1

[∵sin2θ+cos2θ=1]

⇒−cos2θ−4sin2θ+4sinθ.cosθ=−4

⇒4sin2θ+cos2θ−4sinθ.cosθ=4

⇒(2sinθ−cosθ)2=4

[∵a2+b2−2ab=(a−b)2]

⇒2sinθ−cosθ=2
Here θ is used instead of A