[tex]$\frac{\cos\left ( \frac{\pi}{2}-\frac{a}{4} \right )-\sin \left ( \frac{\pi}{2}-\frac{a}{4} \right )\mathrm{tg}\frac{a}{8}}{\sin \left ( \frac{5\pi}{2}+\frac{a}{4} \right )-\sin \left ( \frac{a}{4}-3\pi \right )\mathrm{tg}\frac{a}{8}}=\frac{\sin \frac{a}{4}-\cos \frac{a}{4}\mathrm{tg}\frac{a}{8}}{\cos \frac{a}{4}+\sin \frac{a}{4}\mathrm{tg}\frac{a}{8}}=\frac{\sin \frac{a}{4}-\cos \frac{a}{4}\cdot \frac{\sin a/4}{\cos a/4+1}}{\cos \frac{a}{4}+\sin \frac{a}{4}\cdot \frac{\sin a/4}{\cos a/4+1}}$[/tex]

Знаменатель

[tex]$\cos \frac{a}{4}+\sin \frac{a}{4}\cdot \frac{\sin a/4}{\cos a/4+1}=\frac{\cos^2 \frac{a}{4}+\cos \frac{a}{4}+\sin^2\frac{a}{4}}{\cos \frac{a}{4}+1}=\frac{1+\cos \frac{a}{4}}{\cos \frac{a}{4}+1}=1$[/tex]

Числитель

[tex]$\sin \frac{a}{4}-\cos \frac{a}{4}\cdot \frac{\sin a/4}{\cos a/4+1}=\frac{\sin \frac{a}{4}\cos \frac{a}{4}+\sin \frac{a}{4}-\sin\frac{a}{4}\cos \frac{a}{4}}{\cos \frac{a}{4}+1}=\frac{\sin \frac{a}{4}}{\cos \frac{a}{4}+1}=\mathrm{tg}\frac{a}{8}$[/tex]