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Question

If x = a sinθ + bcosθ and y = acosθ – bsinθ, prove that x2+y2=a2+b2.

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Solution

Given: x=asinθ+bcosθSquaring both sides, we get: x2=a2sin2θ+2absinθcosθ+b2cos2θ ...(i)Also, y=acosθbsinθSquaring both sides, we get:y2=a2cos2θ2absinθcosθ+b2sin2θ ...(ii)LHS= x2+ y2=a2sin2θ+2absinθcosθ+b2cos2θ+ a2cos2θ2absinθcosθ+b2sin2θ [using (i)and (ii)]=a2(sin2θ+cos2θ)+b2(sin2θ+cos2θ)=a2+b2 [sin2θ+cos2θ=1]=RHSHence proved.

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