If `5tantheta=4` , then `(5sintheta-3costheta)/(5sintheta+2costheta)` is equal to

Options

A. `(2)/(3)`
B. `(1)/(4)`
C. `(1)/(6)`
D. `(1)/(3)`

Answer

To solve the problem, we start with the given equation:

1. **Given**: \( 5 \tan \theta = 4 \)

From this, we can express \( \tan \theta \) as:
\[ \tan \theta = \frac{4}{5} \]

2. **Using the identity**:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
We can write:
\[ \frac{\sin \theta}{\cos \theta} = \frac{4}{5} \]
This implies:
\[ 5 \sin \theta = 4 \cos \theta \]

3. **Finding \( \sin \theta \) and \( \cos \theta \)**:
From the equation \( 5 \sin \theta = 4 \cos \theta \), we can express \( \sin \theta \) in terms of \( \cos \theta \):
\[ \sin \theta = \frac{4}{5} \cos \theta \]

4. **Substituting into the expression**:
We need to evaluate:
\[ \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} \]
Substituting \( \sin \theta = \frac{4}{5} \cos \theta \):
\[ 5 \sin \theta = 5 \left(\frac{4}{5} \cos \theta\right) = 4 \cos \theta \]
Now substituting this into the expression:
\[ \frac{4 \cos \theta - 3 \cos \theta}{4 \cos \theta + 2 \cos \theta} \]

5. **Simplifying the expression**:
This simplifies to:
\[ \frac{(4 - 3) \cos \theta}{(4 + 2) \cos \theta} = \frac{1 \cos \theta}{6 \cos \theta} \]
Since \( \cos \theta \) is not zero, we can cancel \( \cos \theta \):
\[ = \frac{1}{6} \]

Thus, the final answer is:
\[ \frac{1}{6} \]