If `cos theta=(4)/(5) and 0lt theta lt 90^(@)`, then the value of `(3cos theta+"2 cosec "theta)/(4sin theta-cot theta)` is :

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the values of sin(theta) and cos(theta)
Given that \( \cos \theta = \frac{4}{5} \), we can use the Pythagorean identity to find \( \sin \theta \).

Using the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the value of \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{16}{25} = 1 \]
\[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 2: Calculate cot(theta)
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting the known values:
\[ \cot \theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

### Step 3: Substitute values into the expression
We need to evaluate:
\[ \frac{3 \cos \theta + 2 \csc \theta}{4 \sin \theta - \cot \theta} \]
First, we need to find \( \csc \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \]

Now substituting the values:
\[ \frac{3 \cdot \frac{4}{5} + 2 \cdot \frac{5}{3}}{4 \cdot \frac{3}{5} - \frac{4}{3}} \]

### Step 4: Simplify the numerator
Calculating the numerator:
\[ 3 \cdot \frac{4}{5} = \frac{12}{5} \]
\[ 2 \cdot \frac{5}{3} = \frac{10}{3} \]
Finding a common denominator (15):
\[ \frac{12}{5} = \frac{36}{15}, \quad \frac{10}{3} = \frac{50}{15} \]
Thus,
\[ \text{Numerator} = \frac{36}{15} + \frac{50}{15} = \frac{86}{15} \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 4 \cdot \frac{3}{5} = \frac{12}{5} \]
Finding a common denominator (15):
\[ \frac{12}{5} = \frac{36}{15}, \quad \frac{4}{3} = \frac{20}{15} \]
Thus,
\[ \text{Denominator} = \frac{36}{15} - \frac{20}{15} = \frac{16}{15} \]

### Step 6: Final calculation
Now we can substitute back into the expression:
\[ \frac{\frac{86}{15}}{\frac{16}{15}} = \frac{86}{16} = \frac{43}{8} \]

### Final Answer:
The value of the expression is:
\[ \frac{43}{8} \]

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