Sintheta +2Costheta= 1 then prove that 2sintheta-costheta=2

sin ϴ + 2 cos ϴ = 1

Squaring both the sides

(sin ϴ + 2 cos ϴ) ² = (1) ²

sin² ϴ + 4 cos² ϴ + 4 sin ϴ cos ϴ = 1

because sin² ϴ = 1 - cos² ϴ & cos² ϴ=1- sin² ϴ

So replacing sin² ϴ by 1 - cos² ϴ and cos² ϴ by 1- sin² ϴ

we get

1 - cos² ϴ + 4 ( 1 - sin² ϴ ) + 4sin ϴ cos ϴ = 1

1 - cos² ϴ + 4 – 4sin² ϴ + 4 sin ϴ cos ϴ = 1

5 – 1 = cos² ϴ +4sin² ϴ - 4 sin ϴ cos ϴ

or

( cos ϴ – 2 sin ϴ ) ² = 4

cos ϴ -2sin ϴ = ± 2 or simply 2 ignoring -2