Question

If acosθ−bsinθ=x & asinθ+bcosθ=y thenprovethat a2 +b2 =x2 +y2

Answer 1

Answer:

a cosθ - b sinθ = x and a sinθ + b cosθ = y

R.H.S. = x2 + y2

= (a cosθ - b sinθ)2 + (a sinθ + b cosθ)2

= a2cos2θ - 2ab cosθ sinθ + b2sin2θ + a2sin2θ + 2absinθ cosθ + b2cos2θ

= (a2+b2) cos2θ + (b2+a2)sin2θ

= (a2+b2)cos2θ + (a2+b2)sin2θ

= (a2+b2)(cos2θ + sin2θ)

= (a2+b2) [∵ cos2θ + sin2θ = 1]

= L.H.S. ∴ a2+b2 = x2+y2.

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