The value of `cot[cos^(-1)(7/25)]` is

Options

A. `25/24`
B. `25/7`
C. `24/25`
D. `7/24`

Explanation

To find the value of \( \cot(\cos^{-1}(\frac{7}{25})) \), we can use a right triangle approach. Here’s a step-by-step solution:

### Step 1: Understand the meaning of \( \cos^{-1}(\frac{7}{25}) \)
Let \( \theta = \cos^{-1}(\frac{7}{25}) \). This means that \( \cos(\theta) = \frac{7}{25} \).

### Step 2: Draw a right triangle
In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side (base) to the hypotenuse. Here, we can set:
- Adjacent side (base) = 7
- Hypotenuse = 25

### Step 3: Use the Pythagorean theorem to find the opposite side
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + \text{Perpendicular}^2 \]
Calculating the squares:
\[ 625 = 49 + \text{Perpendicular}^2 \]
Now, isolate the perpendicular:
\[ \text{Perpendicular}^2 = 625 - 49 = 576 \]
Taking the square root:
\[ \text{Perpendicular} = \sqrt{576} = 24 \]

### Step 4: Find \( \cot(\theta) \)
The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side:
\[ \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{7}{24} \]

### Step 5: Conclusion
Thus, the value of \( \cot(\cos^{-1}(\frac{7}{25})) \) is:
\[ \boxed{\frac{7}{24}} \]

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