If `cos alpha+secalpha = (sqrt(3)) then the value of `cos^(3) alpha+sec^(3)alpha` is

Options

A. 2
B. 1
C. 0
D. 4

Answer

To solve the problem, we need to find the value of \( \cos^3 \alpha + \sec^3 \alpha \) given that \( \cos \alpha + \sec \alpha = \sqrt{3} \).

### Step-by-Step Solution:

1. **Given Equation**:
We start with the equation:
\[ \cos \alpha + \sec \alpha = \sqrt{3} \]

2. **Cubing Both Sides**:
We can cube both sides of the equation:
\[ (\cos \alpha + \sec \alpha)^3 = (\sqrt{3})^3 \]
This simplifies to:
\[ \cos^3 \alpha + \sec^3 \alpha + 3 \cos \alpha \sec \alpha (\cos \alpha + \sec \alpha) = 3\sqrt{3} \]

3. **Substituting Known Values**:
We know that \( \cos \alpha + \sec \alpha = \sqrt{3} \). Therefore, we can substitute this into the equation:
\[ \cos^3 \alpha + \sec^3 \alpha + 3 \cos \alpha \sec \alpha \cdot \sqrt{3} = 3\sqrt{3} \]

4. **Finding \( \cos \alpha \sec \alpha \)**:
Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \). Thus:
\[ \cos \alpha \sec \alpha = 1 \]

5. **Substituting \( \cos \alpha \sec \alpha \)**:
Now we can substitute \( \cos \alpha \sec \alpha = 1 \) into the equation:
\[ \cos^3 \alpha + \sec^3 \alpha + 3 \cdot 1 \cdot \sqrt{3} = 3\sqrt{3} \]
This simplifies to:
\[ \cos^3 \alpha + \sec^3 \alpha + 3\sqrt{3} = 3\sqrt{3} \]

6. **Isolating \( \cos^3 \alpha + \sec^3 \alpha \)**:
We can now isolate \( \cos^3 \alpha + \sec^3 \alpha \):
\[ \cos^3 \alpha + \sec^3 \alpha = 3\sqrt{3} - 3\sqrt{3} \]
Thus:
\[ \cos^3 \alpha + \sec^3 \alpha = 0 \]

### Final Answer:
The value of \( \cos^3 \alpha + \sec^3 \alpha \) is \( 0 \).