if `sin A = 4/5`, then find the value of (4 + tan A) (2 + cosA)

Answer

To solve the problem, we need to find the value of \( (4 + \tan A)(2 + \cos A) \) given that \( \sin A = \frac{4}{5} \).

### Step 1: Find the values of \( \cos A \) and \( \tan A \)

1. **Use the Pythagorean identity**:
\[ \sin^2 A + \cos^2 A = 1 \]
Given \( \sin A = \frac{4}{5} \), we calculate \( \sin^2 A \):
\[ \sin^2 A = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
Now substitute into the identity:
\[ \frac{16}{25} + \cos^2 A = 1 \]
Rearranging gives:
\[ \cos^2 A = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \cos A = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

2. **Calculate \( \tan A \)**:
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

### Step 2: Substitute \( \tan A \) and \( \cos A \) into the expression

Now we substitute \( \tan A \) and \( \cos A \) into the expression \( (4 + \tan A)(2 + \cos A) \):
\[ (4 + \tan A)(2 + \cos A) = \left(4 + \frac{4}{3}\right)\left(2 + \frac{3}{5}\right) \]

### Step 3: Simplify each part

1. **Simplify \( 4 + \frac{4}{3} \)**:
\[ 4 = \frac{12}{3} \quad \text{(to have a common denominator)} \]
So,
\[ 4 + \frac{4}{3} = \frac{12}{3} + \frac{4}{3} = \frac{16}{3} \]

2. **Simplify \( 2 + \frac{3}{5} \)**:
\[ 2 = \frac{10}{5} \quad \text{(to have a common denominator)} \]
So,
\[ 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} \]

### Step 4: Multiply the results

Now we multiply the two simplified results:
\[ (4 + \tan A)(2 + \cos A) = \left(\frac{16}{3}\right)\left(\frac{13}{5}\right) \]
Calculating this gives:
\[ = \frac{16 \times 13}{3 \times 5} = \frac{208}{15} \]

### Final Answer

Thus, the value of \( (4 + \tan A)(2 + \cos A) \) is:
\[ \frac{208}{15} \]

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