If ` DeltaABC` right angle at B, BC = 5 cm and AC -AB =1 cm. then ` (1 + sin C)/(cos C)` is equal to

Options

A. 5
B. 4
C. `1/5`
D. `1/4`

Explanation

To solve the problem, we need to find the value of \((1 + \sin C) / \cos C\) given that triangle \(ABC\) is a right triangle at \(B\), \(BC = 5\) cm, and \(AC - AB = 1\) cm.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \(AB = x\) cm.
- Then, from the given information, \(AC = AB + 1 = x + 1\) cm.
- We know \(BC = 5\) cm.

2. **Apply the Pythagorean theorem**:
- Since triangle \(ABC\) is a right triangle at \(B\), we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values we have:
\[ (x + 1)^2 = x^2 + 5^2 \]

3. **Expand and simplify the equation**:
- Expanding the left side:
\[ x^2 + 2x + 1 = x^2 + 25 \]
- Now, subtract \(x^2\) from both sides:
\[ 2x + 1 = 25 \]

4. **Solve for \(x\)**:
- Rearranging gives:
\[ 2x = 25 - 1 \]
\[ 2x = 24 \]
\[ x = 12 \]

5. **Find \(AB\) and \(AC\)**:
- Now we can find:
\[ AB = x = 12 \text{ cm} \]
\[ AC = x + 1 = 12 + 1 = 13 \text{ cm} \]

6. **Calculate \(\sin C\) and \(\cos C\)**:
- In triangle \(ABC\):
- \(\sin C = \frac{AB}{AC} = \frac{12}{13}\)
- \(\cos C = \frac{BC}{AC} = \frac{5}{13}\)

7. **Substitute into the expression**:
- Now substitute \(\sin C\) and \(\cos C\) into the expression \(\frac{1 + \sin C}{\cos C}\):
\[ \frac{1 + \frac{12}{13}}{\frac{5}{13}} = \frac{\frac{13}{13} + \frac{12}{13}}{\frac{5}{13}} = \frac{\frac{25}{13}}{\frac{5}{13}} \]

8. **Simplify the expression**:
- This simplifies to:
\[ \frac{25}{5} = 5 \]

### Final Answer:
Thus, \((1 + \sin C) / \cos C = 5\).