If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m^2+n^2=a^2+b^2 - Mathematics

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Sum

If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m2+n2=a2+b2

 

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Solution

We have,

`RHS = m^2 + n^2`

`= (acosθ + bsinθ)^2 + (asinθ – bcosθ)^2`

`= (a^2 cos2θ + b^2 sin2θ + 2ab cosθsinθ) + (a^2 sin2θ + b^2 cos2θ – 2ab sinθcosθ)`

`= a^2 (cos^2 θ + sin^2 θ) + b^2 (sin^2 θ + cos^2 θ)`

`= a^2 + b^2 = LHS.`

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