10.29 Derive the formulas expressing $\cos 5 \alpha$ in terms of $\cos \alpha$ and $\sin 5 \alpha$ in terms of $\sin \alpha$.

Perform the operations (10.30-10.32):


10.29 Instruction. Find the degree $(\cos \alpha+i \sin \alpha)^{5}$ in two ways: using de Moivre's formula and using the binomial formula of Newton.

Answer: $\cos 5 \alpha=16 \cos ^{5} \alpha-20 \cos ^{3} \alpha+5 \cos \alpha$, $\sin 5 \alpha=16 \sin ^{5} \alpha-20 \sin ^{3} \alpha+5 \sin \alpha$.