If `7sinalpha=24cosalpha,0ltalphalt(pi)/(2)` then the value of `14tanalpha-75cosalpha-7secalpha` is equal to

Options

A. 3
B. 4
C. 1
D. 2

Answer

To solve the problem step by step, we start with the given equation and work through the calculations systematically.

### Step 1: Start with the given equation
We are given:
\[ 7 \sin \alpha = 24 \cos \alpha \]

### Step 2: Divide both sides by \( \cos \alpha \)
To express \( \tan \alpha \), we divide both sides by \( \cos \alpha \):
\[ \frac{7 \sin \alpha}{\cos \alpha} = 24 \]
This simplifies to:
\[ 7 \tan \alpha = 24 \]

### Step 3: Solve for \( \tan \alpha \)
Now, we can isolate \( \tan \alpha \):
\[ \tan \alpha = \frac{24}{7} \]

### Step 4: Use the Pythagorean theorem to find the hypotenuse
Since \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} \), we can consider:
- Opposite side = 24
- Adjacent side = 7

Using the Pythagorean theorem:
\[ h^2 = 24^2 + 7^2 \]
Calculating:
\[ h^2 = 576 + 49 = 625 \]
Taking the square root gives:
\[ h = 25 \]

### Step 5: Find \( \cos \alpha \) and \( \sec \alpha \)
Now we can find \( \cos \alpha \) and \( \sec \alpha \):
\[ \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{25} \]
\[ \sec \alpha = \frac{1}{\cos \alpha} = \frac{25}{7} \]

### Step 6: Substitute into the expression \( 14 \tan \alpha - 75 \cos \alpha - 7 \sec \alpha \)
Now we substitute these values into the expression:
\[ 14 \tan \alpha - 75 \cos \alpha - 7 \sec \alpha \]
Substituting the values:
\[ = 14 \left(\frac{24}{7}\right) - 75 \left(\frac{7}{25}\right) - 7 \left(\frac{25}{7}\right) \]

### Step 7: Simplify each term
Calculating each term:
1. \( 14 \left(\frac{24}{7}\right) = 48 \)
2. \( 75 \left(\frac{7}{25}\right) = 21 \)
3. \( 7 \left(\frac{25}{7}\right) = 25 \)

### Step 8: Combine the results
Now we combine these results:
\[ 48 - 21 - 25 = 48 - 46 = 2 \]

### Final Answer
Thus, the value of \( 14 \tan \alpha - 75 \cos \alpha - 7 \sec \alpha \) is:
\[ \boxed{2} \]