If ` 7 sin^(2) theta + 3 cos^(2) theta =4 , "then" sec theta + cosec theta ` is equal to

Options

A. `2/sqrt3-2`
B. `2/sqrt3+ 2`
C. `2/sqrt3`
D. None of these

Explanation

To solve the equation \( 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \) and find the value of \( \sec \theta + \csc \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \cos^2 \theta = 1 - \sin^2 \theta \). Substitute this into the equation:

\[ 7 \sin^2 \theta + 3 (1 - \sin^2 \theta) = 4 \]

### Step 2: Simplify the equation
Distributing the 3 gives:

\[ 7 \sin^2 \theta + 3 - 3 \sin^2 \theta = 4 \]

Combine like terms:

\[ (7 - 3) \sin^2 \theta + 3 = 4 \]

This simplifies to:

\[ 4 \sin^2 \theta + 3 = 4 \]

### Step 3: Isolate \( \sin^2 \theta \)
Subtract 3 from both sides:

\[ 4 \sin^2 \theta = 1 \]

Now divide by 4:

\[ \sin^2 \theta = \frac{1}{4} \]

### Step 4: Find \( \sin \theta \)
Taking the square root of both sides gives:

\[ \sin \theta = \frac{1}{2} \]

### Step 5: Determine \( \theta \)
The angle \( \theta \) for which \( \sin \theta = \frac{1}{2} \) is:

\[ \theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ \]

### Step 6: Calculate \( \sec \theta \) and \( \csc \theta \)
Now, we need to find \( \sec \theta \) and \( \csc \theta \):

1. \( \sec \theta = \frac{1}{\cos \theta} \)
2. \( \csc \theta = \frac{1}{\sin \theta} \)

First, find \( \cos \theta \) when \( \sin \theta = \frac{1}{2} \):

\[ \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

Now calculate \( \sec \theta \) and \( \csc \theta \):

\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 7: Add \( \sec \theta \) and \( \csc \theta \)
Now, we can find \( \sec \theta + \csc \theta \):

\[ \sec \theta + \csc \theta = \frac{2}{\sqrt{3}} + 2 \]

### Final Step: Combine the terms
To combine these terms, we can express \( 2 \) with a common denominator:

\[ \sec \theta + \csc \theta = \frac{2}{\sqrt{3}} + \frac{2\sqrt{3}}{\sqrt{3}} = \frac{2 + 2\sqrt{3}}{\sqrt{3}} \]

Thus, the final answer is:

\[ \sec \theta + \csc \theta = \frac{2 + 2\sqrt{3}}{\sqrt{3}} \]