why $ \sin \theta = \frac{7}{8} \cos \theta$? I have an example:
$$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{7}{8} $$
and then this equation is true? why there is cos multiplied?:
$$ \sin \theta = \frac{7}{8} \cos \theta$$

Your initial equation is $$\frac ab = \frac 78.$$ If you multiply this equation by $b$, you get
$$\frac ab \cdot b = \frac 78\cdot b\\
\frac{a\cdot b}{b} = \frac78\cdot b\\
a =\frac78\cdot b$$
