Question

Use De Moivre's theorem to express cos5* in terms of powers of sin* and cos*.

Answer 1

i guess you could take
\[(\cos(\theta)+i\sin(\theta))^5\] and get by demoivre
\[\cos(5\theta)+i\sin(5\theta)\] and also whatever you get when you expand that thing
then equate the real part to \(\cos(5\theta)\)

Answer 2

i have less than no desire to multiply out
\[(\cos(\theta)+i\sin(\theta))^5\] but i can't see any other way to use demoivre