Question

i want the final solution 4: express cos 7 theta in terms of cos theta.. can u help?

Answer 1

using x instead of theta - makes typing easier ;)

cos7x=
cos(3x+4x)=
cos3xcos4x-sin3xsin4x (1)

Now I will first try to do away with cos3x:
(using cos2x=2cos²x-1, sin2x=2sinxcosx and sin²x+cos²x=1)

cos3x=
cos(x+2x)=
cosxcos2x-sinxsin2x=
cosx(2cos²x-1)-sinx2sinxcosx=
cosx(2cos²x-1-2sin²x)=
cosx(2cos²x-1-2(1-cos²x))=
cosx(4cos²x-3).

Now there is only cosx in this part of formula (1).
You see this is already a lot of work.
Now all you have to do is go on and rewrite the other components of (1), using the same techniques!