The maximum value of `cos^(2)A+cos^(2)B-cos^(2)C,` is

Options

A. 0
B. 1
C. 3
D. 2

Answer

To find the maximum value of the expression \( \cos^2 A + \cos^2 B - \cos^2 C \), we can follow these steps:

### Step 1: Understand the Range of Cosine Squared
The cosine function, \( \cos \theta \), varies from -1 to 1. Therefore, when we square it, \( \cos^2 \theta \) will vary from 0 to 1. This means:
\[ 0 \leq \cos^2 A \leq 1, \quad 0 \leq \cos^2 B \leq 1, \quad 0 \leq \cos^2 C \leq 1 \]

### Step 2: Identify Maximum and Minimum Values
To maximize the expression \( \cos^2 A + \cos^2 B - \cos^2 C \), we need to maximize \( \cos^2 A \) and \( \cos^2 B \) while minimizing \( \cos^2 C \).

- The maximum value of \( \cos^2 A \) is 1.
- The maximum value of \( \cos^2 B \) is 1.
- The minimum value of \( \cos^2 C \) is 0.

### Step 3: Substitute Maximum and Minimum Values
Now, we can substitute these values into the expression:
\[ \cos^2 A + \cos^2 B - \cos^2 C = 1 + 1 - 0 \]

### Step 4: Calculate the Result
Calculating the above expression gives:
\[ 1 + 1 - 0 = 2 \]

### Conclusion
Thus, the maximum value of \( \cos^2 A + \cos^2 B - \cos^2 C \) is:
\[ \boxed{2} \]