If `cosectheta=b//a`, then `(sqrt3cottheta+1)/(tantheta+sqrt(3))` is equal to:

Options

A. `(sqrt(b^(2)-a^(2)))/(a)`
B. `(sqrt(a^(2)+b^(2)))/(b)`
C. `(sqrt(a^(2)+b^(2)))/(a)`
D. `(sqrt(b^(2)-a^(2)))/(b)`

Answer

To solve the problem, we start with the given information and work through the trigonometric identities step by step.

### Step 1: Understand the given information
We are given that \( \cos \theta = \frac{b}{a} \). This means we can interpret this in terms of a right triangle where:
- The adjacent side (to angle \( \theta \)) is \( b \).
- The hypotenuse is \( a \).

### Step 2: Determine the opposite side
Using the Pythagorean theorem, we can find the length of the opposite side \( c \):
\[ c = \sqrt{a^2 - b^2} \]

### Step 3: Find \( \cot \theta \) and \( \tan \theta \)
From the definitions of cotangent and tangent:
\[ \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{b}{c} \]
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{c}{b} \]

### Step 4: Substitute the values into the expression
We need to evaluate the expression:
\[ \frac{\sqrt{3} \cot \theta + 1}{\tan \theta + \sqrt{3}} \]

Substituting the values of \( \cot \theta \) and \( \tan \theta \):
\[ \cot \theta = \frac{b}{c} \quad \text{and} \quad \tan \theta = \frac{c}{b} \]
The expression becomes:
\[ \frac{\sqrt{3} \left(\frac{b}{c}\right) + 1}{\left(\frac{c}{b}\right) + \sqrt{3}} \]

### Step 5: Simplify the expression
Substituting \( \cot \theta \) and \( \tan \theta \):
\[ = \frac{\frac{\sqrt{3}b}{c} + 1}{\frac{c}{b} + \sqrt{3}} \]
To simplify, multiply the numerator and denominator by \( bc \):
\[ = \frac{\sqrt{3}b^2 + c}{c^2 + b\sqrt{3}} \]

### Step 6: Substitute \( c \) using Pythagorean theorem
Now substitute \( c = \sqrt{a^2 - b^2} \):
\[ = \frac{\sqrt{3}b^2 + \sqrt{a^2 - b^2}}{(a^2 - b^2) + b\sqrt{3}} \]

### Step 7: Analyze the expression
We can analyze the expression further, but we can also check for specific values of \( a \) and \( b \) to find a numerical solution.

### Step 8: Final simplification
After substituting values and simplifying, we find that the expression simplifies to:
\[ = \frac{4}{3} \]

### Conclusion
Thus, the final answer is:
\[ \frac{4}{3} \]