If cosA + cosB `=4sin^(2)(C/2)`, then

Options

A. `2sinB = sinA + sinC`
B. `cos C = 1-r/R`
C. `cosA + cosB = (2r)/R`
D. a,c,b are in A.P.

Answer

To solve the equation \( \cos A + \cos B = 4 \sin^2\left(\frac{C}{2}\right) \), we will use trigonometric identities and properties of triangles.

### Step-by-step Solution:

1. **Start with the given equation:**
\[ \cos A + \cos B = 4 \sin^2\left(\frac{C}{2}\right) \]

2. **Use the sum-to-product identities:**
The identity for the sum of cosines is:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
So we can rewrite the left-hand side:
\[ 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) = 4 \sin^2\left(\frac{C}{2}\right) \]

3. **Substitute \( A + B \) using the triangle property:**
In a triangle, \( A + B + C = \pi \) (or \( 180^\circ \)), hence:
\[ A + B = \pi - C \]
Therefore,
\[ \frac{A + B}{2} = \frac{\pi - C}{2} = \frac{\pi}{2} - \frac{C}{2} \]

4. **Substitute this back into the equation:**
Now we have:
\[ 2 \cos\left(\frac{\pi}{2} - \frac{C}{2}\right) \cos\left(\frac{A - B}{2}\right) = 4 \sin^2\left(\frac{C}{2}\right) \]
Using the identity \( \cos\left(\frac{\pi}{2} - x\right) = \sin x \):
\[ 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{A - B}{2}\right) = 4 \sin^2\left(\frac{C}{2}\right) \]

5. **Simplify the equation:**
Divide both sides by \( 2 \sin\left(\frac{C}{2}\right) \) (assuming \( \sin\left(\frac{C}{2}\right) \neq 0 \)):
\[ \cos\left(\frac{A - B}{2}\right) = 2 \sin\left(\frac{C}{2}\right) \]

6. **Use the double angle identity:**
Recall that \( \sin C = 2 \sin\left(\frac{C}{2}\right) \cos\left(\frac{C}{2}\right) \). Therefore, we can express \( 2 \sin\left(\frac{C}{2}\right) \) in terms of \( C \):
\[ \cos\left(\frac{A - B}{2}\right) = \sin C \]

7. **Final relationships:**
We can express this in terms of the sides of the triangle:
\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = k \]
where \( a, b, c \) are the sides opposite angles \( A, B, C \) respectively.

### Conclusion:
From the above steps, we can conclude that the relationship holds true under the conditions of a triangle, leading us to the conclusion that \( a, b, c \) are in proportion.