For `0 lt A lt (pi)/(2)`, the value of
`log_(1/2)((1)/(1+2 cos^(2)A)+(2)/(sec^(2)A+2))` is equal to

Answer

To solve the problem, we need to simplify the expression given in the logarithm step by step.

We are given:
\[ \log_{1/2}\left(\frac{1}{1 + 2 \cos^2 A} + \frac{2}{\sec^2 A + 2}\right) \]

### Step 1: Rewrite \(\sec^2 A\)
Recall that \(\sec^2 A = \frac{1}{\cos^2 A}\). Therefore, we can rewrite the second term:
\[ \frac{2}{\sec^2 A + 2} = \frac{2}{\frac{1}{\cos^2 A} + 2} = \frac{2 \cos^2 A}{1 + 2 \cos^2 A} \]

### Step 2: Combine the two fractions
Now we can combine the two fractions:
\[ \frac{1}{1 + 2 \cos^2 A} + \frac{2 \cos^2 A}{1 + 2 \cos^2 A} = \frac{1 + 2 \cos^2 A}{1 + 2 \cos^2 A} = 1 \]

### Step 3: Substitute back into the logarithm
Now we substitute back into the logarithm:
\[ \log_{1/2}(1) \]

### Step 4: Evaluate the logarithm
We know that the logarithm of 1 with any base is 0:
\[ \log_{1/2}(1) = 0 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{0} \]