If `cottheta=3/4`, then the value of `sqrt((1+costheta)/(1-costheta))` is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem where \( \cot \theta = \frac{3}{4} \) and we need to find the value of \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \), we can follow these steps:

### Step 1: Understand the given information
We know that \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{3}{4} \). This means we can represent the sides of a right triangle where the adjacent side is 3 and the opposite side is 4.

### Step 2: Find the hypotenuse
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
\[ \text{hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
\[ \text{hypotenuse} = \sqrt{25} = 5 \]

### Step 3: Calculate \( \cos \theta \)
The cosine of an angle in a right triangle is given by:
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5} \]

### Step 4: Substitute \( \cos \theta \) into the expression
Now we substitute \( \cos \theta \) into the expression \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \):
\[ \sqrt{\frac{1 + \frac{3}{5}}{1 - \frac{3}{5}}} \]

### Step 5: Simplify the expression
First, simplify the numerator and denominator:
\[ 1 + \frac{3}{5} = \frac{5}{5} + \frac{3}{5} = \frac{8}{5} \]
\[ 1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5} \]

Now substitute these values back into the expression:
\[ \sqrt{\frac{\frac{8}{5}}{\frac{2}{5}}} = \sqrt{\frac{8}{5} \cdot \frac{5}{2}} = \sqrt{\frac{8 \cdot 5}{5 \cdot 2}} = \sqrt{\frac{8}{2}} = \sqrt{4} \]

### Step 6: Final calculation
\[ \sqrt{4} = 2 \]

Thus, the value of \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \) is \( 2 \).

### Final Answer:
The value is \( 2 \).

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