If `Tan A = (15)/(8)` and `Tan B = (7)/(24)`, then Tan (A + B) = ?
A. `(416)/(87)`
B. `(87)/(416)`
C. `(304)/(297)`
D. `(297)/(304)`

Options

A. C
B. A
C. B
D. D

Answer

To find \( \tan(A + B) \) given \( \tan A = \frac{15}{8} \) and \( \tan B = \frac{7}{24} \), we can use the formula for the tangent of the sum of two angles:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step 1: Substitute the values of \( \tan A \) and \( \tan B \)

Substituting the given values into the formula:

\[ \tan(A + B) = \frac{\frac{15}{8} + \frac{7}{24}}{1 - \left(\frac{15}{8} \cdot \frac{7}{24}\right)} \]

### Step 2: Find a common denominator for the numerator

To add \( \frac{15}{8} \) and \( \frac{7}{24} \), we need a common denominator. The least common multiple of 8 and 24 is 24.

Convert \( \frac{15}{8} \) to have a denominator of 24:

\[ \frac{15}{8} = \frac{15 \times 3}{8 \times 3} = \frac{45}{24} \]

Now we can add:

\[ \frac{45}{24} + \frac{7}{24} = \frac{45 + 7}{24} = \frac{52}{24} \]

### Step 3: Calculate the denominator

Now, calculate \( 1 - \left(\frac{15}{8} \cdot \frac{7}{24}\right) \):

First, calculate \( \frac{15}{8} \cdot \frac{7}{24} \):

\[ \frac{15 \cdot 7}{8 \cdot 24} = \frac{105}{192} \]

Now, we need to find \( 1 - \frac{105}{192} \):

Convert 1 to a fraction with a denominator of 192:

\[ 1 = \frac{192}{192} \]

So,

\[ 1 - \frac{105}{192} = \frac{192 - 105}{192} = \frac{87}{192} \]

### Step 4: Substitute back into the formula

Now substitute back into the formula for \( \tan(A + B) \):

\[ \tan(A + B) = \frac{\frac{52}{24}}{\frac{87}{192}} \]

### Step 5: Simplify the expression

To divide by a fraction, multiply by its reciprocal:

\[ \tan(A + B) = \frac{52}{24} \cdot \frac{192}{87} \]

Now simplify:

\[ = \frac{52 \cdot 192}{24 \cdot 87} \]

### Step 6: Simplify the fractions

First, simplify \( \frac{192}{24} = 8 \):

\[ = \frac{52 \cdot 8}{87} = \frac{416}{87} \]

### Final Answer

Thus, the value of \( \tan(A + B) \) is:

\[ \boxed{\frac{416}{87}} \]