Question

If 2costheta-sintheta=x and costheta-3sintheta=y,prove that 2x2+y2-2xy=5.

Answer 1

Answer

2cosθ-sinθ=x

or, x²=4cos²θ-4sinθcosθ+sin²θ

or, x²=3cos²θ- and

cosθ-3sinθ=y

or, y²=cos²θ-6sinθcosθ+9sin²θ

xy=(2cosθ-sinθ)(cosθ-3sinθ)

or, xy=2cos²θ-sinθcosθ-6sinθcosθ+3sin²θ

or, xy=2cos²θ-7sinθcosθ+3sin²θ

∴, 2x²+y²-2xy

=2(4cos²θ-4sinθcosθ+sin²θ)+cos²θ-6sinθcosθ+9sin²θ-2(2cos²θ-7sinθcosθ+3sin²θ)

=8cos²θ-8sinθcosθ+2sin²θ+cos²θ-6sinθcosθ+9sin²θ-4cos²θ+14sinθcosθ-6sin²θ

=5cos²θ+5sin²θ

=5(sin²θ+cos²θ)

=5 (Proved) [∵, sin²θ+cos²θ=1]

Answer 2

Hey mate here is your answer :-

2cosθ-sinθ=x

or, x²=3cos²θ- and

xy=(2cosθ-sinθ)(cosθ-3sinθ)

or, xy=2cos²θ-7sinθcosθ+3sin²θ

∴ 2x²+y²-2xy

= 2(4cos²θ-4sinθcosθ +sin²θ)+cos²θ-6sinθcosθ+9sin²θ-2(2cos²θ-7sinθcosθ+3sin²θ)

= 8cos²θ-8sinθcosθ+2sin²θ+cos²θ-6sinθcosθ+9sin²θ-4cos²θ+14sinθcosθ-6sin²θ

= 5cos²θ+5sin²θ

= 5 prooved....

Mark it as brainliest answer....