$\displaystyle \begin{array}{{>{\displaystyle}l}} a.\ 1+\ cosx+cos2x+cos3x\\ =cos^{2} x+sin^{2} x+cos^{2} x-sin^{2} x+2cosxcos2x\\ =2cos^{2} x+2cosxcos2x\\ =2cosx( 1+cos2x)\\ b.\ sinx-sin3x+sin7x-sin5x\\ =2sin4xcos3x-2sin4xcosx\\ =2sin4x( cos3x-cosx)\\ c.\ sinx-sin2x+sin5x+sin8x\\ =2sin3xcos2x+2cos5xsin3x\\ =2sin3x( cos2x+cos5x)\\ d.cos10x-cos8x-cos6x+1\\ =-2sin8xsin2x+cos^{2} 4x+sin^{2} 4x-cos^{2} 4x+sin^{2} 4x\\ =-2sin8xsin2x+2sin^{2} 4x\\ =-4sin4xcos4xsin2x+2sin^{2} 4x\\ =2sin4x( -2cos4xsin2x+sin4x)\\ e.\ cos9x-cos7x+cos3x-cosx\\ =( cos9x-cosx) -( cos7x-cos3x)\\ =-2sin5xsin4x+2sin5xsin2x=0\\ =2sin5x( sin2x-sin4x)\\ f.\ cos7x+sin3x+sin2x-cos3x\\ =( \ cos7x-cos3x) +( sin3x+sin2x)\\ =-2sin5xsin2x+2sin\frac{5x}{2} cos\frac{x}{2}\\ =-8sin\frac{5x}{2} cos\frac{5x}{2} .sin\frac{x}{2} cos\frac{x}{2} +2sin\frac{5x}{2} cos\frac{x}{2}\\ =2sin\frac{5x}{2} cos\frac{x}{2}\left( 1-4scos\frac{5x}{2} .sin\frac{x}{2}\right)\\ \end{array}$
