`Delta ABC` is right angled at B. cot A = 8/15. then what is the value of cos C ?

Options

A. 15/8
B. 44425
C. 17/15
D. 15/17

Answer

To solve the problem step by step, we will use the information given about triangle ABC, which is right-angled at B, and the value of cot A.

### Step 1: Understand the Triangle
Given that triangle ABC is right-angled at B, we can denote:
- AB = a (the side opposite angle C)
- BC = b (the side opposite angle A)
- AC = c (the hypotenuse)

### Step 2: Use the Cotangent Definition
We know that:
\[ \cot A = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{AB}{BC} \]
Given that \(\cot A = \frac{8}{15}\), we can set:
- AB = 8 (adjacent side)
- BC = 15 (opposite side)

### Step 3: Apply the Pythagorean Theorem
To find the hypotenuse AC (c), we apply the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values we have:
\[ AC^2 = 8^2 + 15^2 \]
\[ AC^2 = 64 + 225 \]
\[ AC^2 = 289 \]
Taking the square root:
\[ AC = \sqrt{289} = 17 \]

### Step 4: Calculate Cosine of Angle C
Now, we need to find \(\cos C\):
\[ \cos C = \frac{\text{Adjacent to C}}{\text{Hypotenuse}} = \frac{BC}{AC} \]
Substituting the values:
\[ \cos C = \frac{15}{17} \]

### Final Answer
Thus, the value of \(\cos C\) is \(\frac{15}{17}\).

---