Let a,b,c be the sides of a triangle ABC, a=2c,cos(A-C)+cos B=1. then the value of C is

Options

A. `pi//6`
B. `pi//3`
C. `2pi//3`
D. `5pi//6`

Explanation

To solve the problem, we need to find the value of angle \( C \) given the conditions \( a = 2c \) and \( \cos(A - C) + \cos B = 1 \).

### Step-by-Step Solution:

1. **Write down the given conditions:**
We know that:
- \( a = 2c \)
- \( \cos(A - C) + \cos B = 1 \)

2. **Use the angle sum property of triangles:**
The angles in a triangle sum up to \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Thus, we can express \( B \) in terms of \( A \) and \( C \):
\[ B = 180^\circ - A - C \]

3. **Substitute \( B \) into the cosine equation:**
Using the cosine of the angle difference:
\[ \cos B = \cos(180^\circ - A - C) = -\cos(A + C) \]
Substitute this into the equation:
\[ \cos(A - C) - \cos(A + C) = 1 \]

4. **Use the cosine subtraction and addition formulas:**
The cosine subtraction and addition formulas are:
\[ \cos(A - C) = \cos A \cos C + \sin A \sin C \]
\[ \cos(A + C) = \cos A \cos C - \sin A \sin C \]
Substitute these into the equation:
\[ (\cos A \cos C + \sin A \sin C) - (\cos A \cos C - \sin A \sin C) = 1 \]
Simplifying gives:
\[ 2 \sin A \sin C = 1 \]

5. **Express \( A \) in terms of \( C \):**
From the sine rule:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
Since \( a = 2c \):
\[ \frac{2c}{\sin A} = \frac{c}{\sin C} \]
This simplifies to:
\[ 2 \sin C = \sin A \quad \text{(1)} \]

6. **Substitute \( \sin A \) into the equation:**
Substitute equation (1) into \( 2 \sin A \sin C = 1 \):
\[ 2(2 \sin C) \sin C = 1 \]
This simplifies to:
\[ 4 \sin^2 C = 1 \]
Thus:
\[ \sin^2 C = \frac{1}{4} \]

7. **Find the values of \( C \):**
Taking the square root gives:
\[ \sin C = \frac{1}{2} \]
The angles for which \( \sin C = \frac{1}{2} \) are:
\[ C = 30^\circ \quad \text{or} \quad C = 150^\circ \]

8. **Determine the valid angle for a triangle:**
Since \( C \) must be an angle in a triangle, we discard \( 150^\circ \) (as it would make the sum of angles exceed \( 180^\circ \)). Therefore:
\[ C = 30^\circ \]

### Final Answer:
The value of \( C \) is \( 30^\circ \).