If a b cos A-asin A and y = a cos A + b sin A, then prove that x²+y²= a²+b²
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a2+b2 = x2+y2
Step-by-step explanation:
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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