Question

write cos8x in terms of cosx​

Answer 1

Expand Using DeMoivre's Theorem cos(8x)

cos(8x)

A good method to expand cos(8x) is by using De Moivre's theorem (r(cos(x)+i⋅sin(x))n=rn(cos(nx)+i⋅sin(nx))). When r=1, cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n.

cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n

Expand the right hand side of cos(nx)+i⋅sin(nx)=(cos(x)+i⋅sin(x))n using the binomial theorem.

Expand: (cos(x)+i⋅sin(x))8

Use the Binomial Theorem.

cos8(x)+8cos7(x)(isin(x))+28cos6(x)(isin(x))2+56cos5(x)(isin(x))3+70cos4(x)(isin(x))4+56cos3(x)(isin(x))5+28cos2(x)(isin(x))6+8cos(x)(isin(x))7+(isin(x))8

Simplify terms.

cos8(x)+8icos7(x)sin(x)-28cos6(x)sin2(x)-56icos5(x)sin3(x)+70cos4(x)sin4(x)+56icos3(x)sin5(x)-28cos2(x)sin6(x)-8icos(x)sin7(x)+sin8(x)

Take out the expressions with the imaginary part, which are equal to cos(8x). Remove the imaginary number i.

cos(8x)=cos8(x)-28cos6(x)sin2(x)+70cos4(x)sin4(x)-28cos2(x)sin6(x)+sin8