Step 1: Understanding the Concept:
We need to find the maximum value of a product of two cosines, given a relation between their angles.
We can convert this product into a single trigonometric function to easily find its maximum.
Step 2: Key Formula or Approach:
Use the given relation $A + B = \frac{\pi}{2}$ to express $B$ in terms of $A$.
Use the complementary angle identity: $\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$.
Use the double angle formula: $\sin(2\theta) = 2\sin\theta\cos\theta$.
The maximum value of $\sin(x)$ is $1$.
Step 3: Detailed Explanation:
Let the given expression be $y = \cos A \cdot \cos B$.
We are given that $A + B = \frac{\pi}{2}$, which means $B = \frac{\pi}{2} - A$.
Substitute $B$ into the expression:
\[ y = \cos A \cdot \cos\left(\frac{\pi}{2} - A\right) \]
Using the identity $\cos(\frac{\pi}{2} - A) = \sin A$, we get:
\[ y = \cos A \cdot \sin A \]
Multiply and divide by 2 to use the double angle formula:
\[ y = \frac{1}{2} (2 \sin A \cos A) \]
\[ y = \frac{1}{2} \sin(2A) \]
We know that the sine function oscillates between $-1$ and $1$.
Therefore, the maximum value of $\sin(2A)$ is $1$ (which occurs when $2A = \frac{\pi}{2} \Rightarrow A = \frac{\pi}{4}$).
So, the maximum value of $y$ is:
\[ y_{\max} = \frac{1}{2} \times 1 = \frac{1}{2} \]
Step 4: Final Answer:
The maximum value is $\frac{1}{2}$.