From Förberedande kurs i matematik 1

It seems natural to try to use the addition formula on the numerator of the left-hand side:

\displaystyle \begin{align} & \frac{\cos \left( u+v \right)}{\cos u\cos v}=\frac{\cos u\centerdot \cos v-\sin u\centerdot \sin v}{\cos u\centerdot \cos v} \\ & =1-\frac{\sin u\centerdot \sin v}{\cos u\centerdot \cos v}=1-\tan u\centerdot \tan v. \\ \end{align}