[[Double-or-Half Angle Identities]] $\begin{aligned} & \sin (2 x)=2 \cos x \sin x \\ & \sin (3 x)=3 \cos ^2 x \sin x-\sin ^3 x = 3\sin x-4\sin^3 x\\ & \sin (4 x)=4 \cos ^3 x \sin x-4 \cos x \sin ^3 x \\ & \sin (5 x)=5 \cos ^4 x \sin x-10 \cos ^2 x \sin ^3 x+\sin ^5 x \\ & \cos (2 x)=\cos ^2 x-\sin ^2 x =1-2 \sin ^2 x = -1+2 \cos ^2 x\\ & \cos (3 x)=\cos ^3 x-3 \cos x \sin ^2 x = \cos x\left(1-4 \sin ^2 x\right)=-3 \cos x+4 \cos ^3 x \\ & \cos (4 x)=\cos ^4 x-6 \cos ^2 x \sin ^2 x+\sin ^4 x=1-8 \sin ^2 x+8 \sin ^4 x \\ & \cos (5 x)=\cos ^5 x-10 \cos ^3 x \sin ^2 x+5 \cos x \sin ^4 x=\cos x\left(1-12 \sin ^2 x+16 \sin ^4 x\right)= 5 \cos x-20 \cos ^3 x+16 \cos ^5 x \end{aligned}$