If `cos theta=(4)/(5)` , then value of `(cosec theta)/(1+cot theta)` is

Options

A. `(7)/(5)`
B. `(2)/(7)`
C. `(5)/(7)`
D. `(4)/(7)`

Answer

To solve the problem, we need to find the value of \(\frac{\csc \theta}{1 + \cot \theta}\) given that \(\cos \theta = \frac{4}{5}\).

### Step 1: Determine the sides of the triangle
Since \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{4}{5}\), we can consider a right triangle where:
- Base (adjacent side) = 4
- Hypotenuse = 5

To find the perpendicular side (opposite side), we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \text{Perpendicular}^2 \]
\[ 25 = 16 + \text{Perpendicular}^2 \]
\[ \text{Perpendicular}^2 = 25 - 16 = 9 \]
\[ \text{Perpendicular} = \sqrt{9} = 3 \]

### Step 2: Calculate \(\csc \theta\)
The cosecant function is defined as:
\[ \csc \theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{5}{3} \]

### Step 3: Calculate \(\cot \theta\)
The cotangent function is defined as:
\[ \cot \theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{4}{3} \]

### Step 4: Substitute into the expression
Now, we need to substitute \(\csc \theta\) and \(\cot \theta\) into the expression \(\frac{\csc \theta}{1 + \cot \theta}\):
\[ 1 + \cot \theta = 1 + \frac{4}{3} = \frac{3}{3} + \frac{4}{3} = \frac{7}{3} \]

### Step 5: Final calculation
Now we can substitute these values into the expression:
\[ \frac{\csc \theta}{1 + \cot \theta} = \frac{\frac{5}{3}}{\frac{7}{3}} = \frac{5}{3} \times \frac{3}{7} = \frac{5}{7} \]

Thus, the final answer is:
\[ \frac{\csc \theta}{1 + \cot \theta} = \frac{5}{7} \]