Question

expand sin x and cos x in power of x by maclaurin's theorem​

Answer 1

Answer:

We can prove the expansion of circular functions by using indeterminate coefficients and repeated differentiation. First let’s assign sinx the infinite sequence

sinx=A+Bx+Cx2+Dx3+Ex4+⋯

Differentiating gives the new sequence

cosx=B+2Cx+3Dx2+4Ex3+5Fx4+⋯

So

−sinx=2C+6Dx+12Ex2+20Fx3+30Gx4+⋯

Setting x=0 in the first two equations gives A=0 and B=1. Comparing the coefficients of the two expansions for sinx gives

A=−2CB=−6DC=−12ED=−20F⟹C=0⟹D=−16⟹E=0⟹F=1120

And so on. Therefore

sinx=∑m≥0(−1)mx2m+1(2m+1)!

A similar thing can be done for cosx.