If `7 sin^(2) 0 - cos^(2) 0 + 2 sin 0 = 2, 0^(@) lt 0 lt 90^(@)`, then the value of `(sec 20 + cot 20)/(cosec 20 + tan 20)` is:

Options

A. `(2)/(5)(1+sqrt(3))`
B. `(1)/(5)(1+2sqrt(3))`
C. `(2sqrt(3)+1)/3`
D. 1

Answer

To solve the given problem step by step, we will start with the equation provided and work through to find the value of the expression.

### Step 1: Solve the equation
We start with the equation:
\[ 7 \sin^2 \theta - \cos^2 \theta + 2 \sin \theta = 2 \]

Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we can rewrite the equation:
\[ 7 \sin^2 \theta - (1 - \sin^2 \theta) + 2 \sin \theta = 2 \]
This simplifies to:
\[ 7 \sin^2 \theta + \sin^2 \theta + 2 \sin \theta - 1 = 2 \]
Combining like terms gives:
\[ 8 \sin^2 \theta + 2 \sin \theta - 1 = 2 \]
Now, we move everything to one side:
\[ 8 \sin^2 \theta + 2 \sin \theta - 3 = 0 \]

### Step 2: Use the quadratic formula
This is a quadratic equation in terms of \(\sin \theta\). We can use the quadratic formula:
\[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 8\), \(b = 2\), and \(c = -3\).

Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 8 \cdot (-3) = 4 + 96 = 100 \]

Now applying the quadratic formula:
\[ \sin \theta = \frac{-2 \pm \sqrt{100}}{2 \cdot 8} = \frac{-2 \pm 10}{16} \]
This gives us two possible solutions:
1. \(\sin \theta = \frac{8}{16} = \frac{1}{2}\)
2. \(\sin \theta = \frac{-12}{16} = -\frac{3}{4}\) (not valid since \(\sin \theta\) must be between -1 and 1)

Thus, we have:
\[ \sin \theta = \frac{1}{2} \]
This implies:
\[ \theta = 30^\circ \]

### Step 3: Calculate the expression
Now we need to find the value of:
\[ \frac{\sec 2\theta + \cot 2\theta}{\csc 2\theta + \tan 2\theta} \]

Since \(\theta = 30^\circ\), we have:
\[ 2\theta = 60^\circ \]

Calculating each trigonometric function:
- \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2\)
- \(\cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}}\)
- \(\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\)
- \(\tan 60^\circ = \sqrt{3}\)

Substituting these values into the expression:
\[ \frac{2 + \frac{1}{\sqrt{3}}}{\frac{2}{\sqrt{3}} + \sqrt{3}} \]

### Step 4: Simplify the expression
Finding a common denominator for the numerator:
\[ \frac{2\sqrt{3} + 1}{\sqrt{3}} \]

For the denominator:
\[ \frac{2 + 3}{\sqrt{3}} = \frac{5}{\sqrt{3}} \]

Now we have:
\[ \frac{\frac{2\sqrt{3} + 1}{\sqrt{3}}}{\frac{5}{\sqrt{3}}} = \frac{2\sqrt{3} + 1}{5} \]

### Final Result
Thus, the value of the expression is:
\[ \frac{2\sqrt{3} + 1}{5} \]