Task B-3.3. (20 points) Calculate the sum

$$
\frac{\sin 1}{\cos 0 \cdot \cos 1}+\frac{\sin 1}{\cos 1 \cdot \cos 2}+\frac{\sin 1}{\cos 2 \cdot \cos 3}+\ldots+\frac{\sin 1}{\cos 2008 \cdot \cos 2009}
$$
Solution. Observe the $n$-th term of the expression $\frac{\sin 1}{\cos (n-1) \cos n}$ and rearrange it

$$
\begin{aligned}
\frac{\sin 1}{\cos (n-1) \cos n} & =\frac{\sin [n-(n-1)]}{\cos (n-1) \cos n} \\
& =\frac{\sin n \cos (n-1)}{\cos (n-1) \cos n}-\frac{\cos n \sin (n-1)}{\cos (n-1) \cos n} \\
& =\tan n-\tan(n-1)
\end{aligned}
$$

Now the sum becomes

$$
\begin{aligned}
& \tan 1-\tan 0+\tan 2-\tan 1+\ldots+\tan 2008-\tan 2007+\tan 2009-\tan 2008 \\
= & \tan 2009-\tan 0=\tan 2009
\end{aligned}
$$