If ` sintheta=sqrt3costheta,theta^(@)ltthetalt90^(@)` then the value of `2sin^(2) theta + sec^(2) theta + sin theta sec theta + cosec theta` is ?

Options

A. `(33+10sqrt(3))/(6)`
B. `(19+20sqrt3)/(6)`
C. `(33+10sqrt3)/(3)`
D. `(19+10sqrt3)/(3)`

Answer

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

1. **Given Equation**:
\[ \sin \theta = \sqrt{3} \cos \theta \]

2. **Dividing both sides by \(\cos \theta\)**:
\[ \frac{\sin \theta}{\cos \theta} = \sqrt{3} \]
This implies:
\[ \tan \theta = \sqrt{3} \]

3. **Finding the angle \(\theta\)**:
Since \(\tan \theta = \sqrt{3}\), we know:
\[ \theta = 60^\circ \]
(as \(\tan 60^\circ = \sqrt{3}\) and \(\theta\) is between \(0^\circ\) and \(90^\circ\)).

4. **Calculating the required expression**:
We need to find:
\[ 2 \sin^2 \theta + \sec^2 \theta + \sin \theta \sec \theta + \csc \theta \]

5. **Calculating each term**:
- **Calculate \(\sin 60^\circ\)**:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
- **Calculate \(\cos 60^\circ\)**:
\[ \cos 60^\circ = \frac{1}{2} \]
- **Calculate \(\sec 60^\circ\)**:
\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = 2 \]
- **Calculate \(\csc 60^\circ\)**:
\[ \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{2}{\sqrt{3}} \]

6. **Substituting values into the expression**:
- **Calculating \(2 \sin^2 60^\circ\)**:
\[ 2 \left(\frac{\sqrt{3}}{2}\right)^2 = 2 \cdot \frac{3}{4} = \frac{3}{2} \]
- **Calculating \(\sec^2 60^\circ\)**:
\[ \sec^2 60^\circ = (2)^2 = 4 \]
- **Calculating \(\sin 60^\circ \sec 60^\circ\)**:
\[ \sin 60^\circ \sec 60^\circ = \left(\frac{\sqrt{3}}{2}\right) \cdot 2 = \sqrt{3} \]

7. **Putting it all together**:
Now substituting everything back into the expression:
\[ 2 \sin^2 60^\circ + \sec^2 60^\circ + \sin 60^\circ \sec 60^\circ + \csc 60^\circ = \frac{3}{2} + 4 + \sqrt{3} + \frac{2}{\sqrt{3}} \]

8. **Finding a common denominator**:
The common denominator is \(2\sqrt{3}\):
\[ = \frac{3\sqrt{3}}{2\sqrt{3}} + \frac{8\sqrt{3}}{2\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{3\sqrt{3} + 8\sqrt{3} + 4}{2\sqrt{3}} = \frac{11\sqrt{3}}{2\sqrt{3}} = \frac{11}{2} \]

Thus, the final answer is:
\[ \frac{11}{2} \]