6. Zadatak

Pogodnim grupiranjem i primjenom formula pretvorbi pojednostavnimo bazu na lijevoj strani jednadžbe.

$$\begin{aligned} \frac{\sin1+\sin3+\sin5+\sin7}{\cos1+\cos3+\cos5+\cos7} &=\frac{(\sin1+\sin7)+(\sin3+\sin5)} {(\cos1+\cos7)+(\cos3+\cos5)}\\ &=\frac{2\sin4\cos3+2\sin4\cos1} {2\cos4\cos3+2\cos4\cos1}\\ &=\frac{2\sin4(\cos3+\cos1)} {2\cos4(\cos3+\cos1)} =\operatorname{tg}4. \end{aligned}$$

Dana jednadžba prelazi u

$$(\operatorname{tg}4)^{\frac1{19x}-107}=\frac1{\operatorname{tg}4}.$$

Slijedi

$$\frac1{19x}-107=-1,$$ $$\frac1{19x}=106,$$ $$2014x=1,$$ $$x=\frac1{2014}.$$