🚀 go-pugleaf

Could anyone one please help me with this investigation into multiple angle
formulae i'm doing?  Any assistance would be greatly apprechiated.

The information provided to me was:

;sin 2x = 2.sin x.cos x
          = sin x (cos x)

;sin 3x = sin x (4cos^2 x -1)

and,     cos 2x = cos^2 x - sin ^2x
                      = 2. cos^2 x - 1
                    
       ;   cos 3x = 4.cos^3 x - 3.cos x 


By using an extremly tedious method of expansion, re-arranging and
substitution i managed to re-write further positive integer values of n for
sin nx and cos nx.

For example,
  
sin 4x = sin x (8cos^3 x - 4cos x)

cos 4x = 8cos^4 x - 8cos^2 x + 1)

As the values of n increase, so too does the level of complexity of its
expansion.

I' ve established certain patterns and rules for the expansion in terms of 
sin nx or cos nx. But am at a loss to find a diffinitive link (or rule)
between
the two. 

I have for example discovered that the rule to determine the first term of
the expansion is ;   2^n-1 = first term. ie. 2^(4-1) = 8     (as above)

Would anyone please be able to shed some light on the above problem for me,
as i have reached a mental block on finding a totally perfect pattern or
rule to decide the expansion from simply sin nx or cos nx.


T.I.A,    Al