Prove

$L.H.S =( \mathrm{cosx} +\mathrm{cosy}) +( \mathrm{cosz} + \cos (x+y+z\left)\right)$$= 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right) + 2 \cos \left(\frac{x+y+2z}{2}\right)\cos \left(\frac{x+y}{2}\right)$$[ \mathrm{using} \mathrm{the} \mathrm{formula} \mathrm{cosC} + \mathrm{cosD} = 2\cos \left(\frac{C+D}{2}\right)\cos \left(\frac{C-D}{2}\right)$$\mathrm{and} \cos (-\theta ) = \cos \theta ]$$= 2\cos \left(\frac{x+y}{2}\right)\left[\cos \left(\frac{x-y}{2}\right)+\cos \left(\frac{x+y+2z}{2}\right)\right]$$=2 \cos \left(\frac{x+y}{2}\right)\left[2\cos \left(\frac{x+z}{2}\right)\cos \left(\frac{y+z}{2}\right)\right] \mathrm{using} \mathrm{above} \mathrm{result} \mathrm{again}.$$= 4\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{y+z}{2}\right)\cos \left(\frac{x+z}{2}\right)$$=R.H.S$">