If 7 sin A = 24 cos A then 14 tan A + 25 cos A - 7 sec A equals

Options

A. 0
B. 1
C. 30
D. 32

Answer

To solve the equation \( 14 \tan A + 25 \cos A - 7 \sec A \) given that \( 7 \sin A = 24 \cos A \), we will follow these steps:

### Step 1: Express \(\tan A\) in terms of \(\sin A\) and \(\cos A\)
From the given equation \( 7 \sin A = 24 \cos A \), we can express \(\tan A\) as follows:
\[ \tan A = \frac{\sin A}{\cos A} = \frac{24}{7} \]

### Step 2: Find \(\cos A\) and \(\sin A\)
Using the relationship \( \sin^2 A + \cos^2 A = 1 \):
Let \(\sin A = 24k\) and \(\cos A = 7k\) for some \(k\).
Then:
\[ (24k)^2 + (7k)^2 = 1 \]
\[ 576k^2 + 49k^2 = 1 \]
\[ 625k^2 = 1 \implies k^2 = \frac{1}{625} \implies k = \frac{1}{25} \]
Thus,
\[ \sin A = 24k = \frac{24}{25}, \quad \cos A = 7k = \frac{7}{25} \]

### Step 3: Calculate \(\sec A\)
The secant function is the reciprocal of the cosine function:
\[ \sec A = \frac{1}{\cos A} = \frac{1}{\frac{7}{25}} = \frac{25}{7} \]

### Step 4: Substitute values into the expression
Now substitute \(\tan A\), \(\cos A\), and \(\sec A\) into the expression \(14 \tan A + 25 \cos A - 7 \sec A\):
\[ 14 \tan A = 14 \times \frac{24}{7} = 48 \]
\[ 25 \cos A = 25 \times \frac{7}{25} = 7 \]
\[ 7 \sec A = 7 \times \frac{25}{7} = 25 \]

### Step 5: Combine the results
Now we can combine these results:
\[ 14 \tan A + 25 \cos A - 7 \sec A = 48 + 7 - 25 \]
\[ = 55 - 25 = 30 \]

### Final Answer
Thus, the value of \( 14 \tan A + 25 \cos A - 7 \sec A \) is \( \boxed{30} \).

---