Question

If acosx-bsinx =c, prove that asinx + bcosx = +- root(a 2 +b 2 -c 2 )

Answer 1

acosx - bsinx = c ------(1)
take square both sides
a²cos²x + b²sin²x -2absinx.cosx = c² ----(1)

asinx + bcosx = L --------(2)
take square both sides
a²sin²x + b²cos²x +2absinx.cosx = L² ----(2)


add equations (1)and (2)
a²(sin²x + cos²x ) + b²( sin²x + cos²x ) = c² + L²

a² + b² = c² + L²

L² = a² + b² -c²

L = ±√( a² + b² - c²)

asinx + bcosx =±√(a² + b² - c²)

hence proved ///