If `costheta=4/5` and `0ltthetalt90^@`, then the value of `(3costheta+2cosectheta)/(4sintheta-cottheta)` is:

Options

A. `-43/2`
B. `-41/2`
C. `43/8`
D. `-43/6`

Answer

To solve the problem, we need to evaluate the expression \((3 \cos \theta + 2 \csc \theta) / (4 \sin \theta - \cot \theta)\) given that \(\cos \theta = \frac{4}{5}\) and \(0 < \theta < 90^\circ\).

### Step-by-Step Solution:

1. **Identify the values of \(\sin \theta\) and \(\csc \theta\)**:
- We know \(\cos \theta = \frac{4}{5}\).
- Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
- Therefore, \(\sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}\) (since \(\theta\) is in the first quadrant, \(\sin \theta\) is positive).
- The cosecant function is the reciprocal of sine:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \]

2. **Calculate \(\cot \theta\)**:
- The cotangent function is the ratio of cosine to sine:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

3. **Substitute values into the expression**:
- Now substitute \(\cos \theta\), \(\csc \theta\), \(\sin \theta\), and \(\cot \theta\) into the expression:
\[ \frac{3 \cos \theta + 2 \csc \theta}{4 \sin \theta - \cot \theta} = \frac{3 \cdot \frac{4}{5} + 2 \cdot \frac{5}{3}}{4 \cdot \frac{3}{5} - \frac{4}{3}} \]

4. **Simplify the numerator**:
- Calculate the numerator:
\[ 3 \cdot \frac{4}{5} = \frac{12}{5} \]
\[ 2 \cdot \frac{5}{3} = \frac{10}{3} \]
- Find a common denominator (15):
\[ \frac{12}{5} = \frac{36}{15}, \quad \frac{10}{3} = \frac{50}{15} \]
- Thus, the numerator becomes:
\[ \frac{36}{15} + \frac{50}{15} = \frac{86}{15} \]

5. **Simplify the denominator**:
- Calculate the denominator:
\[ 4 \cdot \frac{3}{5} = \frac{12}{5} \]
- Convert \(\frac{12}{5}\) and \(\frac{4}{3}\) to a common denominator (15):
\[ \frac{12}{5} = \frac{36}{15}, \quad \frac{4}{3} = \frac{20}{15} \]
- Thus, the denominator becomes:
\[ \frac{36}{15} - \frac{20}{15} = \frac{16}{15} \]

6. **Combine the results**:
- Now substitute back into the expression:
\[ \frac{\frac{86}{15}}{\frac{16}{15}} = \frac{86}{16} \]

7. **Simplify the fraction**:
- Divide both the numerator and the denominator by 2:
\[ \frac{86 \div 2}{16 \div 2} = \frac{43}{8} \]

### Final Answer:
The value of \(\frac{3 \cos \theta + 2 \csc \theta}{4 \sin \theta - \cot \theta}\) is \(\frac{43}{8}\).