The maximum and minimum values of expression `(4-2costheta)` respectively are

Options

A. 4 and 0
B. 4 and 2
C. 6 and 0
D. 6 and 2

Answer

To find the maximum and minimum values of the expression \( 4 - 2\cos\theta \), we can follow these steps:

### Step 1: Understand the range of \(\cos\theta\)
The cosine function, \(\cos\theta\), varies between -1 and 1. Therefore, we need to consider these extreme values to find the maximum and minimum of the expression.

### Step 2: Find the maximum value of the expression
To find the maximum value of \( y = 4 - 2\cos\theta \), we need to minimize \(\cos\theta\). The minimum value of \(\cos\theta\) is -1.

Substituting \(\cos\theta = -1\) into the expression:
\[ y = 4 - 2(-1) = 4 + 2 = 6 \]
Thus, the maximum value of the expression is 6.

### Step 3: Find the minimum value of the expression
To find the minimum value of \( y = 4 - 2\cos\theta \), we need to maximize \(\cos\theta\). The maximum value of \(\cos\theta\) is 1.

Substituting \(\cos\theta = 1\) into the expression:
\[ y = 4 - 2(1) = 4 - 2 = 2 \]
Thus, the minimum value of the expression is 2.

### Conclusion
The maximum and minimum values of the expression \( 4 - 2\cos\theta \) are:
- Maximum value: 6
- Minimum value: 2

---