Question

If 2cos theta-sin theta=x and cos theta-3 sin theta =y. prove that 2x^+y^-2xy=5

Answer 1

According to the Question

(2 cosθ - sinθ) = x   (cosθ - 3 sinθ) = y

Substitute the values of x and y

2x^2 + y^2 − 2xy (Left Hand Side)

= 2(2 cosθ − sinθ)2 + (cosθ − 3 sinθ)2 − 2(2 cosθ − sinθ)(cosθ − 3 sinθ)

= 2(4cos^2θ − 4cosθ sinθ + sin^2θ) + (cos^2θ − 6cosθ sinθ + 9sin^2θ) - 2(2cos^2θ − 7cosθ sinθ + 3sin^2θ)

= 8cos^2θ − 8cosθ sinθ + 2sin^2θ + cos^2θ − 6cosθ sinθ + 9sin^2θ − 4cos^2θ + 14cosθ sinθ − 6sin^2θ

= 5cos^2θ + 5sin^2θ

= 5(cos^2θ + sin^2θ)

= 5(1) = 5    ⇒   ( cos^2θ + sin^2θ = 1)

LHS = RHS