Sine Meets Cosine

Note that

$\sin^4\left(\frac{\pi}{8}\right)+\cos^4\left(\frac{\pi}{8}\right)$

$=\left[\sin^2\left(\frac{\pi}{8}\right)+\cos^2\left(\frac{\pi}{8}\right)\right]^2-2\sin^2\left(\frac{\pi}{8}\right)\cos^2\left(\frac{\pi}{8}\right)$

$=1-2\sin^2\left(\frac{\pi}{8}\right)\cos^2\left(\frac{\pi}{8}\right)$

$=1-\frac{\left[2\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{\pi}{8}\right)\right]^2}{2}$

$=1-\frac{\left[\sin\left(\frac{\pi}{4}\right)\right]^2}{2}$

$=1-\frac{\left(\frac{1}{\sqrt{2}}\right)^2}{2}$

$=1-\frac{1}{4}=\frac{3}{4}.$

Similarly,

$\sin^4\left(\frac{7\pi}{8}\right)+\cos^4\left(\frac{7\pi}{8}\right)$

$=1-\frac{\left[\sin\left(\frac{7\pi}{4}\right)\right]^2}{2}$

$=1-\frac{\left(-\frac{1}{\sqrt{2}}\right)^2}{2}$

$=\frac{3}{4}.$

Hence, $36x=36 \times \left(\frac{3}{4}+\frac{3}{4}\right)=\boxed{54}$ , and we are done. $_\square$