`cot[cos^(-1)"7/25]=`

Options

A. `25/24`
B. `25/7`
C. `24/25`
D. none of these

Answer

To solve the problem \( \cot(\cos^{-1}(7/25)) \), we can follow these steps:

### Step 1: Understand the expression
We start with \( \theta = \cos^{-1}(7/25) \). This means that \( \cos(\theta) = \frac{7}{25} \).

### Step 2: Draw a right triangle
To visualize this, we can draw a right triangle where:
- The adjacent side (base) is \( 7 \),
- The hypotenuse is \( 25 \).

### Step 3: Find the opposite side
We can use the Pythagorean theorem to find the opposite side:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + \text{opposite}^2 \]
\[ 625 = 49 + \text{opposite}^2 \]
\[ \text{opposite}^2 = 625 - 49 = 576 \]
\[ \text{opposite} = \sqrt{576} = 24 \]

### Step 4: Calculate cotangent
Now that we have the lengths of all sides of the triangle, we can find \( \cot(\theta) \):
\[ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{7}{24} \]

### Step 5: Conclusion
Thus, we have:
\[ \cot(\cos^{-1}(7/25)) = \frac{7}{24} \]

### Final Answer
The final answer is \( \frac{7}{24} \).

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