If sinA = 4/5, then find the value of (1 + tan A)/(1 + cot A)?

Options

A. `7/3`
B. `3/4`
C. `4/7`
D. `4/3`

Answer

To solve the problem, we need to find the value of \((1 + \tan A)/(1 + \cot A)\) given that \(\sin A = \frac{4}{5}\).

### Step-by-step Solution:

1. **Identify the values of sine, cosine, and tangent:**
Given \(\sin A = \frac{4}{5}\), we can find \(\cos A\) using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting the value of \(\sin A\):
\[ \left(\frac{4}{5}\right)^2 + \cos^2 A = 1 \]
\[ \frac{16}{25} + \cos^2 A = 1 \]
\[ \cos^2 A = 1 - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \cos A = \frac{3}{5} \]

2. **Calculate \(\tan A\) and \(\cot A\):**
\(\tan A\) is defined as:
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]
\(\cot A\) is the reciprocal of \(\tan A\):
\[ \cot A = \frac{1}{\tan A} = \frac{3}{4} \]

3. **Substitute \(\tan A\) and \(\cot A\) into the expression:**
We need to find:
\[ \frac{1 + \tan A}{1 + \cot A} = \frac{1 + \frac{4}{3}}{1 + \frac{3}{4}} \]
Simplifying the numerator:
\[ 1 + \frac{4}{3} = \frac{3}{3} + \frac{4}{3} = \frac{7}{3} \]
Simplifying the denominator:
\[ 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \]

4. **Combine the results:**
Now we have:
\[ \frac{\frac{7}{3}}{\frac{7}{4}} = \frac{7}{3} \times \frac{4}{7} \]
The \(7\)s cancel out:
\[ = \frac{4}{3} \]

### Final Answer:
Thus, the value of \(\frac{1 + \tan A}{1 + \cot A}\) is \(\frac{4}{3}\).