P(x)=`(x-cos36^@)(x-cos84^@)(x-cos156^@)` then coefficient of `x^2` is

Options

A. `0`
B. `1`
C. `-(1)/(2)`
D. `(sqrt(5) - 1)/(2)`

Explanation

To find the coefficient of \( x^2 \) in the polynomial \( P(x) = (x - \cos 36^\circ)(x - \cos 84^\circ)(x - \cos 156^\circ) \), we can follow these steps:

### Step 1: Identify the Roots
The roots of the polynomial \( P(x) \) are:
- \( \alpha = \cos 36^\circ \)
- \( \beta = \cos 84^\circ \)
- \( \gamma = \cos 156^\circ \)

### Step 2: Use Vieta's Formulas
According to Vieta's formulas, for a cubic polynomial of the form \( ax^3 + bx^2 + cx + d \), the sum of the roots (which corresponds to the coefficient of \( x^2 \)) is given by:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
Since \( a = 1 \) in our polynomial, we have:
\[ \alpha + \beta + \gamma = -b \]
Thus, to find the coefficient \( b \), we need to calculate \( \alpha + \beta + \gamma \).

### Step 3: Calculate \( \alpha + \beta + \gamma \)
We will calculate:
\[ \alpha + \beta + \gamma = \cos 36^\circ + \cos 84^\circ + \cos 156^\circ \]

### Step 4: Simplify Using Trigonometric Identities
Using the cosine addition formulas:
- \( \cos 84^\circ = \cos(90^\circ - 6^\circ) = \sin 6^\circ \)
- \( \cos 156^\circ = \cos(180^\circ - 24^\circ) = -\cos 24^\circ \)

Thus, we can express the sum as:
\[ \cos 36^\circ + \sin 6^\circ - \cos 24^\circ \]

### Step 5: Use the Cosine Addition Formula
We can also use the cosine addition formulas to combine these terms:
\[ \cos 36^\circ + \cos 84^\circ + \cos 156^\circ = \cos 36^\circ + \cos(90^\circ - 6^\circ) - \cos 24^\circ \]
This can be simplified further using known values or identities.

### Step 6: Evaluate the Sum
After evaluating, we find:
\[ \cos 36^\circ + \cos 84^\circ + \cos 156^\circ = 0 \]

### Step 7: Conclusion
Since \( \alpha + \beta + \gamma = 0 \), we have:
\[ -b = 0 \implies b = 0 \]
Thus, the coefficient of \( x^2 \) in the polynomial \( P(x) \) is \( 0 \).

### Final Answer
The coefficient of \( x^2 \) is \( 0 \).

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