If `3 sin theta + 4 cos theta=5`, then find the value of `4 sin theta-3 cos theta`.

Answer

To solve the equation \(3 \sin \theta + 4 \cos \theta = 5\) and find the value of \(4 \sin \theta - 3 \cos \theta\), we can follow these steps:

### Step 1: Rearrange the given equation
We start with the equation:
\[ 3 \sin \theta + 4 \cos \theta = 5 \]
We can rearrange it to express \(4 \cos \theta\):
\[ 4 \cos \theta = 5 - 3 \sin \theta \]

### Step 2: Square both sides
Next, we square both sides of the equation:
\[ (3 \sin \theta + 4 \cos \theta)^2 = 5^2 \]
Expanding the left side:
\[ (3 \sin \theta)^2 + 2(3 \sin \theta)(4 \cos \theta) + (4 \cos \theta)^2 = 25 \]
This simplifies to:
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 \cos^2 \theta = 25 \]

### Step 3: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can substitute \(16 \cos^2 \theta\) with \(16(1 - \sin^2 \theta)\):
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16(1 - \sin^2 \theta) = 25 \]
This leads to:
\[ 9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 - 16 \sin^2 \theta = 25 \]
Combining like terms gives:
\[ -7 \sin^2 \theta + 24 \sin \theta \cos \theta - 9 = 0 \]

### Step 4: Substitute \( \cos \theta \)
Now, we can express \(\cos \theta\) in terms of \(\sin \theta\):
From the rearranged equation \(4 \cos \theta = 5 - 3 \sin \theta\):
\[ \cos \theta = \frac{5 - 3 \sin \theta}{4} \]

### Step 5: Substitute back into the equation
We can substitute this expression for \(\cos \theta\) back into our equation:
\[ 4 \sin \theta - 3 \left(\frac{5 - 3 \sin \theta}{4}\right) \]
This simplifies to:
\[ 4 \sin \theta - \frac{15 - 9 \sin \theta}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 16 \sin \theta - (15 - 9 \sin \theta) = 0 \]
This simplifies to:
\[ 16 \sin \theta - 15 + 9 \sin \theta = 0 \]
Combining the terms gives:
\[ 25 \sin \theta - 15 = 0 \]

### Step 6: Solve for \(\sin \theta\)
From this, we find:
\[ 25 \sin \theta = 15 \implies \sin \theta = \frac{15}{25} = \frac{3}{5} \]

### Step 7: Find \(\cos \theta\)
Using the Pythagorean identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos \theta = \frac{4}{5} \]

### Step 8: Calculate \(4 \sin \theta - 3 \cos \theta\)
Now we can find \(4 \sin \theta - 3 \cos \theta\):
\[ 4 \sin \theta - 3 \cos \theta = 4 \left(\frac{3}{5}\right) - 3 \left(\frac{4}{5}\right) \]
This simplifies to:
\[ \frac{12}{5} - \frac{12}{5} = 0 \]

### Final Answer
Thus, the value of \(4 \sin \theta - 3 \cos \theta\) is:
\[ \boxed{0} \]