If `cot theta = (3)/(4)` , then `(sin theta - cos theta)/(sin theta + cos theta)` =

Options

A. `(1)/(7)`
B. `(2)/(7)`
C. `(1)/(4)`
D. `(3)/(4)`

Answer

To solve the problem where \( \cot \theta = \frac{3}{4} \) and we need to find the value of \( \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \), we can follow these steps:

### Step 1: Understand the cotangent definition
Given \( \cot \theta = \frac{3}{4} \), we can interpret this in terms of a right triangle. In a right triangle, \( \cot \theta \) is defined as the ratio of the adjacent side (base) to the opposite side (perpendicular).

### Step 2: Assign triangle sides
Let:
- Adjacent side (base) = 3
- Opposite side (perpendicular) = 4

### Step 3: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ h = \sqrt{(\text{perpendicular})^2 + (\text{base})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 4: Find sine and cosine values
Now we can find \( \sin \theta \) and \( \cos \theta \):
\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} = \frac{4}{5} \]
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{3}{5} \]

### Step 5: Substitute sine and cosine into the expression
Now substituting these values into the expression \( \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \):
\[ \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\frac{4}{5} - \frac{3}{5}}{\frac{4}{5} + \frac{3}{5}} \]

### Step 6: Simplify the expression
Calculating the numerator and denominator:
- Numerator:
\[ \frac{4}{5} - \frac{3}{5} = \frac{1}{5} \]
- Denominator:
\[ \frac{4}{5} + \frac{3}{5} = \frac{7}{5} \]

Now substituting back:
\[ \frac{\frac{1}{5}}{\frac{7}{5}} = \frac{1}{5} \times \frac{5}{7} = \frac{1}{7} \]

### Final Answer
Thus, the value of \( \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \) is \( \frac{1}{7} \).

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