Question

If sinΘ+2cosΘ=1,then prove that 2sinΘ-cosΘ=2​.please prove fast

Answer 1

sinθ+2cosθ=1

Squaring both sides,

sin²θ+2.sinθ.2cosθ+4cos²θ=1

or, (1-cos²θ)+4sinθcosθ+4(1-sin²θ)=1 [∵, sin²θ+cos²θ=1]

or, 1-cos²θ+4sinθcosθ+4-4sin²θ=1

or, -4sin²θ+4sinθcosθ-cos²θ=1-1-4

or, -(4sin²θ-4sinθcosθ+cos²θ)=-4

or, (2sinθ)²-2.2sinθcosθ+(cosθ)²=4

or, (2sinθ-cosθ)²=2²

or, 2sinθ-cosθ=2 (Proved)