<s>[INST] Showing that $f(4x) + 4f(2x) =8\cos^4(x)-3$, where $f(x) = \cos x$ 
It is given that 
  $$f(x)= \sin(x+30^\circ) + \cos(x+60^\circ)$$
A) Show that $$f(x)= \cos(x)$$
B) Hence, show that 
  $$f(4x) + 4f(2x) =8\cos^4(x)-3$$

I managed to prove $f(x)$ equals $\cos(x)$, but after that I'm stumped.
 [/INST] $$f(x)=\sin(30^{\circ}+x)+\sin(30^{\circ}-x)=2\sin30^{\circ}\cos{x}=\cos{x}.$$
Thus, $$f(4x)+4f(2x)=\cos4x+4\cos2x=8\cos^4x-8\cos^2x+1+8\cos^2x-4=8\cos^4x-3.$$
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