IF `A+B=pi/2`, the maximum value of cos Acos B is

Options

A. `1/2`
B. `3/4`
C. 1
D. `4/3`

Answer

To find the maximum value of \( \cos A \cos B \) given that \( A + B = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the identity for cosine
Since \( A + B = \frac{\pi}{2} \), we can express \( B \) in terms of \( A \):
\[ B = \frac{\pi}{2} - A \]

### Step 2: Substitute into the expression
Now, substitute \( B \) into the expression \( \cos A \cos B \):
\[ \cos A \cos B = \cos A \cos\left(\frac{\pi}{2} - A\right) \]

### Step 3: Simplify using the co-function identity
Using the co-function identity \( \cos\left(\frac{\pi}{2} - A\right) = \sin A \), we have:
\[ \cos A \cos B = \cos A \sin A \]

### Step 4: Use the double angle identity
We can use the double angle identity for sine, which states that:
\[ \sin 2A = 2 \sin A \cos A \]
Thus,
\[ \cos A \sin A = \frac{1}{2} \sin 2A \]

### Step 5: Determine the maximum value of \( \sin 2A \)
The maximum value of \( \sin 2A \) is 1, since the range of the sine function is from -1 to 1. Therefore:
\[ \max(\cos A \sin A) = \frac{1}{2} \max(\sin 2A) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Conclusion
The maximum value of \( \cos A \cos B \) is:
\[ \frac{1}{2} \]

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