If `7sin^(2)theta+3cos^(2)theta=4` then the value of `sectheta+cosectheta` is

Options

A. `(2)/(sqrt(3))-2`
B. `(2)/(sqrt(3))+2`
C. `(2)/(sqrt(3))`
D. None of these

Answer

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 using the Pythagorean identity
We know that \(\cos^2\theta = 1 - \sin^2\theta\). We can substitute this into the equation:

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

### Step 2: Simplify the equation
Distributing the \(3\) gives us:

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

Combining like terms results in:

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

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: Solve for \(\sin\theta\)
Taking the square root of both sides gives:

\[ \sin\theta = \frac{1}{2} \quad \text{or} \quad \sin\theta = -\frac{1}{2} \]

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

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

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

\[ \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta} \]

For \(\theta = 30^\circ\):

\[ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2} \]

Thus,

\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{2}{\sqrt{3}}, \quad \csc 30^\circ = \frac{1}{\sin 30^\circ} = 2 \]

Now adding these together:

\[ \sec 30^\circ + \csc 30^\circ = \frac{2}{\sqrt{3}} + 2 \]

### Step 7: Simplify the expression
To combine these, we can express \(2\) with a common denominator:

\[ 2 = \frac{2\sqrt{3}}{\sqrt{3}} \]

So,

\[ \sec 30^\circ + \csc 30^\circ = \frac{2}{\sqrt{3}} + \frac{2\sqrt{3}}{\sqrt{3}} = \frac{2 + 2\sqrt{3}}{\sqrt{3}} \]

### Final Answer
Thus, the value of \(\sec\theta + \csc\theta\) is:

\[ \sec\theta + \csc\theta = \frac{2(1 + \sqrt{3})}{\sqrt{3}} \]

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