Q22 of 104 Page 204

acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a2 + b2 = p2 + q2.

acosθ + bsinθ = p (i)

asinθ – bcosθ = q (ii)


Squaring and adding (i) and (ii)


∴ (acosθ + bsinθ)2 + (asinθ – bcosθ)2 = p2 + q2


∴ a2cos2θ + 2abcosθsinθ + b2sin2θ + a2sin2θ – 2absinθcosθ + b2cos2θ = p2 + q2


∴ a2(cos2θ + sin2θ) + b2 (sin2θ + cos2θ) = p2 + q2


∴a2 (1) + b2 (1) = p2 + q2


∴ a2 + b2 = p2 + q2


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