Parte (A):
Notamos que $\cos\frac{7\pi}{8} = -\cos\frac{\pi}{8}$ y $\cos\frac{5\pi}{8} = -\cos\frac{3\pi}{8}$. Al estar a la cuarta, los signos negativos desaparecen.
$E = 2[\cos^4(\frac{\pi}{8}) + \cos^4(\frac{3\pi}{8})]$. Además $\cos\frac{3\pi}{8} = \sin\frac{\pi}{8}$.
$E = 2[\cos^4(\frac{\pi}{8}) + \sin^4(\frac{\pi}{8})] = 2[1 - 2\sin^2(\frac{\pi}{8})\cos^2(\frac{\pi}{8})] = 2[1 - \frac{1}{2}\sin^2(\frac{\pi}{4})] = 2[1 - \frac{1}{2}(\frac{1}{2})] = 2[3/4] = 3/2$. (R)

Parte (B):
Usamos $\sin\theta \sin(60-\theta) \sin(60+\theta) = \frac{1}{4}\sin 3\theta$.
Para $\theta = 12^\circ$: $\sin 12^\circ \sin 48^\circ \sin 72^\circ = \frac{1}{4}\sin 36^\circ$.
Entonces $\sin 12^\circ \sin 48^\circ = \frac{\sin 36^\circ}{4 \sin 72^\circ} = \frac{\sin 36^\circ}{8 \sin 36^\circ \cos 36^\circ} = \frac{1}{8\cos 36^\circ}$.
La expresión es $\frac{\sin 54^\circ}{8\cos 36^\circ}$. Como $\sin 54^\circ = \cos 36^\circ$, el valor es $1/8$. (P)

Parte (C):
$\sin 6^\circ \sin 66^\circ \sin 54^\circ = \frac{1}{4}\sin 18^\circ \rightarrow \sin 6^\circ \sin 66^\circ = \frac{\sin 18^\circ}{4 \sin 54^\circ}$.
$\sin 18^\circ \sin 42^\circ \sin 78^\circ = \frac{1}{4}\sin 54^\circ \rightarrow \sin 42^\circ \sin 78^\circ = \frac{\sin 54^\circ}{4 \sin 18^\circ}$.
Multiplicando: $(\frac{\sin 18^\circ}{4 \sin 54^\circ}) \cdot (\frac{\sin 54^\circ}{4 \sin 18^\circ}) = 1/16$. (S)

Parte (D):
Similarmente, usando $\tan\theta \tan(60-\theta) \tan(60+\theta) = \tan 3\theta$:
$\tan 6^\circ \tan 54^\circ \tan 66^\circ = \tan 18^\circ$
$\tan 18^\circ \tan 42^\circ \tan 78^\circ = \tan 54^\circ$
Multiplicando: $\tan 6^\circ \tan 66^\circ \tan 42^\circ \tan 78^\circ = \frac{\tan 18^\circ}{\tan 54^\circ} \cdot \frac{\tan 54^\circ}{\tan 18^\circ} = 1$. (T)

Resultado:
$$ \boxed{(A)\rightarrow R, (B)\rightarrow P, (C)\rightarrow S, (D)\rightarrow T} $$