It is known that `sinbeta=4/5` and `0 < beta < pi` then the value of `(sqrt3 sin(alpha+beta)-2/cos(pi/6) cos(alpha+beta))/(sin alpha)` is

Options

A. (a)independent of `alpha` for all `beta` in `(0 , pi)`
B. (b)`(5)/(sqrt(3))` for `tan beta gt 1`
C. (c)`(sqrt(3)(7+24 cot alpha))/(15)` for `tan beta lt 0`
D. (d)zero for `tan beta gt 0`

Answer

To solve the problem step by step, we start with the given information and use trigonometric identities to simplify the expression.

### Given:
- \( \sin \beta = \frac{4}{5} \)
- \( 0 < \beta < \pi \)

### Step 1: Find \( \cos \beta \)
Using the Pythagorean identity:
\[ \sin^2 \beta + \cos^2 \beta = 1 \]
Substituting the value of \( \sin \beta \):
\[ \left(\frac{4}{5}\right)^2 + \cos^2 \beta = 1 \]
\[ \frac{16}{25} + \cos^2 \beta = 1 \]
\[ \cos^2 \beta = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ \cos \beta = \pm \frac{3}{5} \]
Since \( 0 < \beta < \pi \), \( \cos \beta \) can be negative in the second quadrant. Therefore:
\[ \cos \beta = -\frac{3}{5} \]

### Step 2: Rewrite the expression
We need to evaluate:
\[ \frac{\sqrt{3} \sin(\alpha + \beta) - 2 \cos\left(\frac{\pi}{6}\right) \cos(\alpha + \beta)}{\sin \alpha} \]

### Step 3: Find \( \cos\left(\frac{\pi}{6}\right) \)
Using the known value:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

### Step 4: Substitute into the expression
Now substituting this value into the expression:
\[ \frac{\sqrt{3} \sin(\alpha + \beta) - 2 \cdot \frac{\sqrt{3}}{2} \cos(\alpha + \beta)}{\sin \alpha} \]
This simplifies to:
\[ \frac{\sqrt{3} \sin(\alpha + \beta) - \sqrt{3} \cos(\alpha + \beta)}{\sin \alpha} \]

### Step 5: Factor out \( \sqrt{3} \)
Factoring out \( \sqrt{3} \):
\[ \frac{\sqrt{3} \left( \sin(\alpha + \beta) - \cos(\alpha + \beta) \right)}{\sin \alpha} \]

### Step 6: Use the angle addition formula
Using the angle addition formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]
Substituting these into the expression gives:
\[ \sin(\alpha + \beta) - \cos(\alpha + \beta) = \left( \sin \alpha \cdot -\frac{3}{5} + \cos \alpha \cdot \frac{4}{5} \right) - \left( \cos \alpha \cdot -\frac{3}{5} - \sin \alpha \cdot \frac{4}{5} \right) \]

### Step 7: Simplify the expression
Combining the terms:
\[ = \left( -\frac{3}{5} \sin \alpha + \frac{4}{5} \cos \alpha + \frac{3}{5} \cos \alpha + \frac{4}{5} \sin \alpha \right) \]
\[ = \left( \frac{1}{5} \sin \alpha + \frac{7}{5} \cos \alpha \right) \]

### Step 8: Final expression
Now substituting this back into our expression:
\[ \frac{\sqrt{3} \left( \frac{1}{5} \sin \alpha + \frac{7}{5} \cos \alpha \right)}{\sin \alpha} \]
This simplifies to:
\[ = \frac{\sqrt{3}}{5} + \frac{7\sqrt{3} \cos \alpha}{5 \sin \alpha} \]

### Conclusion
The final value of the expression is:
\[ \frac{\sqrt{3}}{5} + \frac{7\sqrt{3} \cos \alpha}{5 \sin \alpha} \]