If `sin x = (4)/(5)`, then `("Tan x")/("Cot x") = ?`
A. `(13)/(9)`
B. `(3)/(4)`
C. `(9)/(16)`
D. `(16)/(9)`

Options

A. D
B. A
C. B
D. C

Answer

To solve the problem, we need to find the value of \(\frac{\tan x}{\cot x}\) given that \(\sin x = \frac{4}{5}\).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \(\sin x = \frac{4}{5}\). In a right triangle, this means that the opposite side (perpendicular) to angle \(x\) is 4 units and the hypotenuse is 5 units.

**Hint**: Remember that \(\sin x\) is defined as the ratio of the opposite side to the hypotenuse.

2. **Finding the Base (Adjacent Side)**:
To find the adjacent side (base), we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \text{Adjacent}^2 \]
\[ 25 = 16 + \text{Adjacent}^2 \]
\[ \text{Adjacent}^2 = 25 - 16 = 9 \]
\[ \text{Adjacent} = \sqrt{9} = 3 \]

**Hint**: Use the Pythagorean theorem to find the length of the adjacent side.

3. **Calculating \(\tan x\) and \(\cot x\)**:
- \(\tan x = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{4}{3}\)
- \(\cot x = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{3}{4}\)

**Hint**: Recall that \(\tan x\) is the ratio of the opposite side to the adjacent side, while \(\cot x\) is the reciprocal.

4. **Finding \(\frac{\tan x}{\cot x}\)**:
Now, we can find \(\frac{\tan x}{\cot x}\):
\[ \frac{\tan x}{\cot x} = \frac{\frac{4}{3}}{\frac{3}{4}} = \frac{4}{3} \times \frac{4}{3} = \frac{16}{9} \]

**Hint**: To divide fractions, multiply by the reciprocal of the denominator.

5. **Conclusion**:
Therefore, the value of \(\frac{\tan x}{\cot x}\) is \(\frac{16}{9}\).

**Final Answer**: D. \(\frac{16}{9}\)