Q49 of 268 Page 5

If a cos θ + b sinθ = c, then prove that a sinθ – b cos θ = ±

Let

(a cos θ + b sin θ)2 + (a sin θ – b cos θ)2 = a2cos2θ + b2 sin2θ + 2ab cos θ sin θ + a2sin2θ


+ b2 cos2θ – 2ab cos θ sin θ


⇒ c2 + (a sin θ – b cos θ)2 = a2 (cos2 θ + sin2 θ) + b2 (cos2 θ + sin2 θ)


⇒ c2 + (a sin θ – b cos θ)2 = a2 + b2


⇒ (a sin θ – b cos θ)2 = a2 + b2 – c2


⇒ (a sin θ – b cos θ) = ±√ (a2 + b2 – c2)


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