Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of

$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$
To find the maximum value of the expression 

\[
S = \sin\theta_1\cos\theta_2 + \sin\theta_2\cos\theta_3 + \ldots + \sin\theta_{2007}\cos\theta_{2008} + \sin\theta_{2008}\cos\theta_1,
\]

we can use the Cauchy-Schwarz inequality in the following form:

\[
\left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right).
\]

In our case, let \( a_i = \sin\theta_i \) and \( b_i = \cos\theta_{i+1} \) (with \( \theta_{2009} = \theta_1 \)). Then we have:

\[
S = \sum_{i=1}^{2008} \sin\theta_i \cos\theta_{i+1}.
\]

Applying the Cauchy-Schwarz inequality, we get:

\[
\left( \sum_{i=1}^{2008} \sin\theta_i \cos\theta_{i+1} \right)^2 \leq \left( \sum_{i=1}^{2008} \sin^2\theta_i \right) \left( \sum_{i=1}^{2008} \cos^2\theta_{i+1} \right).
\]

Since \( \sum_{i=1}^{2008} \sin^2\theta_i \leq 2008 \) and \( \sum_{i=1}^{2008} \cos^2\theta_{i+1} \leq 2008 \), we have:

\[
\left( \sum_{i=1}^{2008} \sin\theta_i \cos\theta_{i+1} \right)^2 \leq 2008 \cdot 2008 = 2008^2.
\]

Taking the square root of both sides, we get:

\[
\sum_{i=1}^{2008} \sin\theta_i \cos\theta_{i+1} \leq 2008.
\]

However, we need to find the maximum value of \( S \). To achieve equality in the Cauchy-Schwarz inequality, the sequences \( \sin\theta_i \) and \( \cos\theta_{i+1} \) must be proportional. This means there exists a constant \( k \) such that:

\[
\sin\theta_i = k \cos\theta_{i+1}.
\]

Given that \( \sin^2\theta_i + \cos^2\theta_i = 1 \), we can set \( \theta_i = \frac{\pi}{4} \) for all \( i \). This choice ensures that \( \sin\theta_i = \cos\theta_i = \frac{\sqrt{2}}{2} \). Substituting these values into the original sum, we get:

\[
S = \sum_{i=1}^{2008} \left( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} \right) = \sum_{i=1}^{2008} \frac{1}{2} = 1004.
\]

Thus, the maximum value of the given expression is \( 1004 \).

The final answer is \(\boxed{1004}\).