If `sin x = (3)/(5), 0 le x le 90^(@)`, then the value of `cot x.sec x ` is :

Options

A. `(5)/(3)`
B. `(3)/(5)`
C. `(4)/(5)`
D. `(3)/(4)`

Answer

To solve the problem, we need to find the value of \( \cot x \cdot \sec x \) given that \( \sin x = \frac{3}{5} \) and \( 0 \leq x \leq 90^\circ \).

### Step-by-step Solution:

1. **Find \( \cos x \)**:
We know that \( \sin^2 x + \cos^2 x = 1 \).
Given \( \sin x = \frac{3}{5} \), we can find \( \cos x \) as follows:
\[ \cos^2 x = 1 - \sin^2 x = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Taking the square root, we get:
\[ \cos x = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

2. **Find \( \cot x \)**:
The cotangent function is defined as:
\[ \cot x = \frac{\cos x}{\sin x} \]
Substituting the values we found:
\[ \cot x = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

3. **Find \( \sec x \)**:
The secant function is the reciprocal of the cosine function:
\[ \sec x = \frac{1}{\cos x} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]

4. **Calculate \( \cot x \cdot \sec x \)**:
Now we can find the product:
\[ \cot x \cdot \sec x = \left(\frac{4}{3}\right) \cdot \left(\frac{5}{4}\right) \]
Simplifying this gives:
\[ \cot x \cdot \sec x = \frac{4 \cdot 5}{3 \cdot 4} = \frac{5}{3} \]

### Final Answer:
The value of \( \cot x \cdot \sec x \) is \( \frac{5}{3} \).