\(\cos^{2}(2x)-\sin^{2}(2x)=0\)
\(\rightarrow \sin^{2}(2x)=\cos^{2}(2x)\)
\(\rightarrow \frac{\sin^{2}(2x)}{\cos^{2}(2x)}=1\)
\(\rightarrow \tan^{2}(2x)=1\)
\(\rightarrow \tan(2x)=\pm 1\)
\(2x\) can range anywhere in \([0, 4\pi)\)
So:
\(\rightarrow 2x=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, \frac{13\pi}{4}, \frac{15\pi}{4}\)
\(\rightarrow x=\frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, \frac{9\pi}{8}, \frac{11\pi}{8}, \frac{13\pi}{8}, \frac{15\pi}{8}\)
