Text Simplify the expression cos(π−x)+sin (π2+x)+cos(−x)\cos(\pi-x)+\sin\!\left(\tfrac{\pi}{2}+x\right)+\cos(-x)cos(π−x)+sin(2π+x)+cos(−x). Solution Apply the associated-arc identities: cos(π−x)=−cosx\cos(\pi-x)=-\cos xcos(π−x)=−cosx; sin (π2+x)=cosx\sin\!\left(\tfrac{\pi}{2}+x\right)=\cos xsin(2π+x)=cosx; cos(−x)=cosx\cos(-x)=\cos xcos(−x)=cosx (cosine is even). Adding up: −cosx+cosx+cosx=cosx.-\cos x+\cos x+\cos x=\cos x.−cosx+cosx+cosx=cosx. cosx \boxed{\;\cos x\;}cosx