If a sin theta + b cos theta = c. Prove that a cos theta - b sin theta = root a² + b² - c²

If a sin theta + b cos theta = c. Prove that a cos theta - b sin theta

1.	If a sin theta + b cos theta = c. Prove that a cos theta - b sin theta = root a² + b² - c²
Answer» asinθ + bcosθ = ctaking square both sides,(asinθ + bcosθ)² = c²⇒a²sin²θ + b²cos²θ + 2absinθ.cosθ = c² --------(1)Let acosθ - bsinθ = xSquaring both sides(acosθ - bsinθ)² = x²⇒a²cos²θ + b²sin²θ -2absinθ.cosθ = x² ------(2)Add equation (1) and (2),a²sin²θ + b²cos²θ +2abinθ.cosθ + a²cos²θ + b²sin²θ -2absinθ.cosθ = c² + x²⇒(a² + b²)cos²θ + (a² +b²)sin²θ = c² + x²⇒(a² + b²)[sin²θ + cos²θ ] = c² + x²⇒(a² + b²) = c² + x² [∵ sin²x + cos²x = 1 ]⇒(a² + b² - c²) = x²Take square root both sides,Hence, acosθ - bsinθ =\xa0