Chapter 7 - Section 7.2 - Addition and Subtraction Formulas - 7.2 Exercises - Page 551: 45

$\sin \left( x+y+z\right) =\sin \left( \left( x+y\right) +z\right) =\sin \left( x+y\right) \cos z+\cos \left( x+y\right) \sin z =\left( \sin x\cos y+\cos x\sin y\right) \cos z+\left( \cos x\cos y-\sin x\sin y\right) \sin z=\sin x\cos y\cos z+\cos x\sin y\cos z+\cos x\cos y\sin z-\sin x\sin y\sin z $

$\sin \left( x+y+z\right) =\sin \left( \left( x+y\right) +z\right) =\sin \left( x+y\right) \cos z+\cos \left( x+y\right) \sin z =\left( \sin x\cos y+\cos x\sin y\right) \cos z+\left( \cos x\cos y-\sin x\sin y\right) \sin z=\sin x\cos y\cos z+\cos x\sin y\cos z+\cos x\cos y\sin z-\sin x\sin y\sin z $