If Cot `x = (5)/(12)`, then Sin x `+ (1)/("Cot x") + Sec x = ?`

Options

A. 173/13
B. 77/13
C. 75/13
D. 17/13

Answer

To solve the problem, we need to find the value of the expression \( \sin x + \frac{1}{\cot x} + \sec x \) given that \( \cot x = \frac{5}{12} \).

### Step-by-Step Solution:

1. **Understand the Given Information:**
We know that \( \cot x = \frac{5}{12} \). This means that in a right triangle, the adjacent side (base) is 5 and the opposite side (perpendicular) is 12.

2. **Calculate the Hypotenuse:**
Using the Pythagorean theorem, we can find the hypotenuse \( h \):
\[ h = \sqrt{(\text{adjacent})^2 + (\text{opposite})^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

3. **Find \( \sin x \):**
The sine of angle \( x \) is given by the ratio of the opposite side to the hypotenuse:
\[ \sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13} \]

4. **Find \( \cot x \):**
We already know \( \cot x = \frac{5}{12} \). Therefore, \( \frac{1}{\cot x} \) is:
\[ \frac{1}{\cot x} = \frac{12}{5} \]

5. **Find \( \sec x \):**
The secant of angle \( x \) is given by the ratio of the hypotenuse to the adjacent side:
\[ \sec x = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{13}{5} \]

6. **Combine All Values:**
Now we can substitute these values into the expression:
\[ \sin x + \frac{1}{\cot x} + \sec x = \frac{12}{13} + \frac{12}{5} + \frac{13}{5} \]

7. **Finding a Common Denominator:**
The common denominator for \( 13 \) and \( 5 \) is \( 65 \). We convert each term:
\[ \frac{12}{13} = \frac{12 \times 5}{13 \times 5} = \frac{60}{65} \]
\[ \frac{12}{5} = \frac{12 \times 13}{5 \times 13} = \frac{156}{65} \]
\[ \frac{13}{5} = \frac{13 \times 13}{5 \times 13} = \frac{169}{65} \]

8. **Add the Values Together:**
Now we can add the fractions:
\[ \frac{60}{65} + \frac{156}{65} + \frac{169}{65} = \frac{60 + 156 + 169}{65} = \frac{385}{65} \]

9. **Simplifying the Fraction:**
Now we simplify \( \frac{385}{65} \):
\[ \frac{385 \div 5}{65 \div 5} = \frac{77}{13} \]

### Final Answer:
Thus, the value of \( \sin x + \frac{1}{\cot x} + \sec x \) is \( \frac{77}{13} \).