If 5 tan `theta` =4, find the value of `(5 sin theta - 3 cos theta)/(5 sin theta +2 cos theta)`

Answer

To solve the problem, we need to find the value of the expression \((5 \sin \theta - 3 \cos \theta)/(5 \sin \theta + 2 \cos \theta)\) given that \(5 \tan \theta = 4\).

### Step 1: Express \(\tan \theta\)
From the given equation \(5 \tan \theta = 4\), we can express \(\tan \theta\) as:
\[ \tan \theta = \frac{4}{5} \]

### Step 2: Use the identity for \(\sin \theta\) and \(\cos \theta\)
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus, we can write:
\[ \sin \theta = \tan \theta \cdot \cos \theta = \frac{4}{5} \cos \theta \]

### Step 3: Substitute \(\sin \theta\) in the expression
Now, we substitute \(\sin \theta\) in the expression:
\[ 5 \sin \theta = 5 \left(\frac{4}{5} \cos \theta\right) = 4 \cos \theta \]
So, we can rewrite the expression:
\[ \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} = \frac{4 \cos \theta - 3 \cos \theta}{4 \cos \theta + 2 \cos \theta} \]

### Step 4: Simplify the expression
Now, simplify the numerator and the denominator:
\[ = \frac{(4 - 3) \cos \theta}{(4 + 2) \cos \theta} = \frac{1 \cos \theta}{6 \cos \theta} \]
Since \(\cos \theta \neq 0\), we can cancel \(\cos \theta\):
\[ = \frac{1}{6} \]

### Final Answer
Thus, the value of the expression \(\frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta}\) is:
\[ \frac{1}{6} \]