Which identity to use?

Relevant wiki: Sum and Difference Trigonometric Formulas - Problem Solving

Use $\cos 3x=4\cos^3x-3\cos x$ and rearrange for $\cos^3 x$ and substitute in $\mathfrak F$ :

$\mathfrak F=\small{\dfrac{\cos^{3}{1}}{1}-\dfrac{\cos^{3}{3}}{3}+\dfrac{\cos^{3}{9}}{9}-\dfrac{\cos^{3}{27}}{27}\cdots}$ $\small{=\dfrac 14\left(\dfrac{\cos 3+3\cos 1}{1}-\dfrac{\cos 9+3\cos 3}{3}+\dfrac{\cos 27+3\cos9}{9}-\cdots\right)}$

$\small{=\dfrac 14\left((\cancel{\color{#20A900}{\cos 3}}+3\cos 1)-(\cancel{\color{#D61F06}{\dfrac{\cos 9}3}}+\cancel{\color{#20A900}{\cos 3}})+(\cancel{\color{#3D99F6}{\dfrac{\cos 27}{9}}}+\cancel{\color{#D61F06}{\dfrac{\cos 9}3}})-\cdots\right)}$

$\large =\dfrac{3\cos 1}4$

$a=3,b=4,c=d=1$

$\large a+b+c+d=\boxed{9}$