If `2cos^(2)theta-5costheta+2=0,0^(@)ltthetalt90^(@)` then the value of `(1)/((cosectheta+cottheta))` is :

Options

A. `(sqrt(3))/(3)`
B. `sqrt3`
C. `2sqrt3`
D. `(1)/(3)`

Answer

To solve the equation \( 2\cos^2\theta - 5\cos\theta + 2 = 0 \) and find the value of \( \frac{1}{\csc\theta + \cot\theta} \), we will follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ 2\cos^2\theta - 5\cos\theta + 2 = 0 \]
Let \( x = \cos\theta \). The equation becomes:
\[ 2x^2 - 5x + 2 = 0 \]

### Step 2: Factor the quadratic equation
To factor the quadratic, we look for two numbers that multiply to \( 2 \times 2 = 4 \) and add to \( -5 \). The numbers are \( -4 \) and \( -1 \).
Thus, we can rewrite the equation as:
\[ 2x^2 - 4x - x + 2 = 0 \]
Grouping the terms:
\[ 2x(x - 2) - 1(x - 2) = 0 \]
Factoring out \( (x - 2) \):
\[ (2x - 1)(x - 2) = 0 \]

### Step 3: Find the roots
Setting each factor to zero gives:
1. \( 2x - 1 = 0 \) → \( x = \frac{1}{2} \)
2. \( x - 2 = 0 \) → \( x = 2 \)

Since \( \cos\theta \) must be in the range \([-1, 1]\), we discard \( x = 2 \). Thus, we have:
\[ \cos\theta = \frac{1}{2} \]

### Step 4: Determine the angle
The angle \( \theta \) that satisfies \( \cos\theta = \frac{1}{2} \) in the range \( 0^\circ < \theta < 90^\circ \) is:
\[ \theta = 60^\circ \]

### Step 5: Calculate \( \csc\theta \) and \( \cot\theta \)
Now we need to find \( \csc\theta \) and \( \cot\theta \):
\[ \csc\theta = \frac{1}{\sin\theta} = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
\[ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Step 6: Find \( \csc\theta + \cot\theta \)
Now we can calculate:
\[ \csc\theta + \cot\theta = \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{2 + 1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 7: Calculate \( \frac{1}{\csc\theta + \cot\theta} \)
Finally, we find:
\[ \frac{1}{\csc\theta + \cot\theta} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \frac{1}{\csc\theta + \cot\theta} \) is:
\[ \frac{1}{\sqrt{3}} \]