If `cos alpha and cos beta (0 lt a lt beta lt pi)` are the roots of the quadratic equation `4x^(2) - 3 = 0`. Then what is the value of `sec alpha xx sec beta` ?

Options

A. `-(4)/(3)`
B. `(4)/(3)`
C. `(3)/(4)`
D. `(-(3)/(4))`

Answer

To solve the problem, we start with the quadratic equation given:

**Step 1: Solve the quadratic equation.**
The equation is:
\[ 4x^2 - 3 = 0 \]

To find the roots, we can rearrange the equation:
\[ 4x^2 = 3 \]
\[ x^2 = \frac{3}{4} \]
Taking the square root of both sides, we get:
\[ x = \pm \sqrt{\frac{3}{4}} \]
\[ x = \pm \frac{\sqrt{3}}{2} \]

**Step 2: Identify the roots.**
Given that \( \cos \alpha \) and \( \cos \beta \) are the roots, we can assign:
\[ \cos \alpha = \frac{\sqrt{3}}{2} \]
\[ \cos \beta = -\frac{\sqrt{3}}{2} \]

However, since \( 0 < \alpha < \beta < \pi \), we must have:
\[ \cos \alpha = \frac{\sqrt{3}}{2} \quad \text{(valid for } \alpha = \frac{\pi}{6} \text{)} \]
\[ \cos \beta = -\frac{\sqrt{3}}{2} \quad \text{(valid for } \beta = \frac{5\pi}{6} \text{)} \]

**Step 3: Calculate secant values.**
The secant function is the reciprocal of the cosine function:
\[ \sec \alpha = \frac{1}{\cos \alpha} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
\[ \sec \beta = \frac{1}{\cos \beta} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} \]

**Step 4: Multiply secants.**
Now, we need to find the product \( \sec \alpha \cdot \sec \beta \):
\[ \sec \alpha \cdot \sec \beta = \left( \frac{2}{\sqrt{3}} \right) \cdot \left( -\frac{2}{\sqrt{3}} \right) \]
\[ = -\frac{4}{3} \]

Thus, the value of \( \sec \alpha \cdot \sec \beta \) is:
\[ \boxed{-\frac{4}{3}} \]

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