If `cos x=3/5` and x lies in the fourth quadrant find the values of `cosec x+ cot x`.

Answer

To solve the problem, we need to find the value of \( \csc x + \cot x \) given that \( \cos x = \frac{3}{5} \) and \( x \) lies in the fourth quadrant.

### Step-by-Step Solution:

1. **Identify the values of sine and cosine**:
Given \( \cos x = \frac{3}{5} \), we know that in the fourth quadrant, cosine is positive.

2. **Use the Pythagorean identity**:
We can find \( \sin x \) using the identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting \( \cos x \):
\[ \sin^2 x + \left(\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 x + \frac{9}{25} = 1 \]
\[ \sin^2 x = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

3. **Calculate \( \sin x \)**:
Since \( x \) is in the fourth quadrant, \( \sin x \) will be negative:
\[ \sin x = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \]

4. **Find \( \csc x \)**:
The cosecant function is the reciprocal of sine:
\[ \csc x = \frac{1}{\sin x} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \]

5. **Find \( \cot x \)**:
The cotangent function is the ratio of cosine to sine:
\[ \cot x = \frac{\cos x}{\sin x} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} \]

6. **Calculate \( \csc x + \cot x \)**:
Now we can add \( \csc x \) and \( \cot x \):
\[ \csc x + \cot x = -\frac{5}{4} + \left(-\frac{3}{4}\right) = -\frac{5}{4} - \frac{3}{4} = -\frac{8}{4} = -2 \]

### Final Answer:
\[ \csc x + \cot x = -2 \]