If in a `DeltaABC` , `cos A + 2 cosB+cosC= 2`, then a,b, c are in

Options

A. A.P.
B. H.P.
C. G.P
D. none of these

Answer

To solve the problem, we need to find the relationship between the angles \( A, B, C \) of triangle \( ABC \) given the equation:

\[ \cos A + 2 \cos B + \cos C = 2 \]

### Step-by-Step Solution:

**Step 1: Rearrange the equation.**

We start with the given equation:

\[ \cos A + 2 \cos B + \cos C = 2 \]

We can rearrange it to isolate \( \cos A + \cos C \):

\[ \cos A + \cos C = 2 - 2 \cos B \]

**Hint for Step 1:** Rearranging helps to simplify the equation and focus on the relationship between the angles.

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**Step 2: Use the identity for \( \cos A + \cos C \).**

We can use the trigonometric identity:

\[ \cos A + \cos C = 2 \cos\left(\frac{A + C}{2}\right) \cos\left(\frac{A - C}{2}\right) \]

Substituting this into our rearranged equation gives:

\[ 2 \cos\left(\frac{A + C}{2}\right) \cos\left(\frac{A - C}{2}\right) = 2 - 2 \cos B \]

**Hint for Step 2:** Using trigonometric identities can help express the angles in a different form that may reveal relationships.

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**Step 3: Simplify the equation.**

Dividing both sides by 2, we have:

\[ \cos\left(\frac{A + C}{2}\right) \cos\left(\frac{A - C}{2}\right) = 1 - \cos B \]

We can express \( 1 - \cos B \) using the identity \( 1 - \cos B = 2 \sin^2\left(\frac{B}{2}\right) \):

\[ \cos\left(\frac{A + C}{2}\right) \cos\left(\frac{A - C}{2}\right) = 2 \sin^2\left(\frac{B}{2}\right) \]

**Hint for Step 3:** Transforming the equation into a sine function can help relate the angles more directly.

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**Step 4: Use the sine rule.**

From the sine rule, we know:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]

Thus, we can express \( \sin A \) and \( \sin C \) in terms of \( a, b, c \):

\[ \sin A = \frac{a}{k}, \quad \sin C = \frac{c}{k} \]

Substituting these into the equation gives:

\[ \frac{a}{k} + \frac{c}{k} = 2 \cdot \frac{b}{k} \]

Multiplying through by \( k \):

\[ a + c = 2b \]

**Hint for Step 4:** The sine rule provides a direct relationship between the sides and angles, which can be very useful in triangle problems.

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**Step 5: Conclude the relationship.**

The equation \( a + c = 2b \) indicates that \( a, b, c \) are in Arithmetic Progression (AP).

Thus, we conclude:

\[ \text{The angles } A, B, C \text{ are in AP.} \]

### Final Answer:
The angles \( A, B, C \) are in **Arithmetic Progression (AP)**.

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