If `2costheta-sintheta=x` and `costheta-3sintheta=ydot` Prove that `2x^2+y^2-2x y=5.`

Explanation

Given `(2 costheta - sintheta) = x`
and `(costheta - 3 sintheta) = y`

Put the values of x and y in the equation `2x^2 + y^2 − 2xy `(LHS)


=` 2(2 costheta − sintheta)^2 + (costheta − 3 sintheta)^2 − 2(2 costheta − sintheta)(costheta − 3 sintheta)`

= `2(4cos^2theta − 4costheta sintheta + sin^2theta) + (cos^2theta − 6costheta sintheta + 9sin^2theta) − 2(2cos^2theta − 7costheta sintheta + 3sin^2theta)`

= `8cos^2theta − 8costheta sintheta + 2sin^2theta + cos^2theta − 6costheta sintheta + 9sin^2theta − 4cos^2theta + 14costheta sintheta − 6sin^2theta`

= `5cos^2theta + 5sin^2theta`

= `5(cos^2theta + sin^2theta)`

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