If `cosA=7/25`, find the value of `tanA+cotA`

Answer

To solve the problem where \( \cos A = \frac{7}{25} \) and we need to find the value of \( \tan A + \cot A \), we can follow these steps:

### Step 1: Understand the relationship of cosine in a right triangle
Given \( \cos A = \frac{7}{25} \), we can interpret this in the context of a right triangle. Here, the cosine of angle \( A \) is defined as the ratio of the adjacent side (base) to the hypotenuse.

### Step 2: Identify the sides of the triangle
From \( \cos A = \frac{7}{25} \):
- Let the adjacent side (base) = 7
- Let the hypotenuse = 25

### Step 3: Use the Pythagorean theorem to find the opposite side (perpendicular)
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + \text{perpendicular}^2 \]
\[ 625 = 49 + \text{perpendicular}^2 \]
\[ \text{perpendicular}^2 = 625 - 49 = 576 \]
\[ \text{perpendicular} = \sqrt{576} = 24 \]

### Step 4: Calculate \( \tan A \) and \( \cot A \)
Now we can find \( \tan A \) and \( \cot A \):
- \( \tan A = \frac{\text{perpendicular}}{\text{base}} = \frac{24}{7} \)
- \( \cot A = \frac{\text{base}}{\text{perpendicular}} = \frac{7}{24} \)

### Step 5: Find \( \tan A + \cot A \)
Now, we can add \( \tan A \) and \( \cot A \):
\[ \tan A + \cot A = \frac{24}{7} + \frac{7}{24} \]

### Step 6: Find a common denominator and simplify
The common denominator for \( 7 \) and \( 24 \) is \( 168 \):
\[ \tan A + \cot A = \frac{24 \times 24}{168} + \frac{7 \times 7}{168} = \frac{576 + 49}{168} = \frac{625}{168} \]

### Final Answer
Thus, the value of \( \tan A + \cot A \) is:
\[ \frac{625}{168} \]

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