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

Options

A. `31^(2)`
B. `30^(2)`
C. `17^(2)`
D. `7^(2)`

Answer

To solve the problem, we start with the given information that \( \tan \theta = \frac{5}{4} \). We need to find the value of

\[ \left( \frac{3 \sin \theta + 4 \cos \theta}{3 \sin \theta - 4 \cos \theta} \right)^2. \]

### Step 1: Rewrite the expression using tangent
We can express \( \sin \theta \) and \( \cos \theta \) in terms of \( \tan \theta \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can write:

\[ \sin \theta = \tan \theta \cdot \cos \theta. \]

Substituting \( \tan \theta = \frac{5}{4} \), we have:

\[ \sin \theta = \frac{5}{4} \cos \theta. \]

### Step 2: Substitute into the expression
Now we substitute \( \sin \theta \) in the expression:

\[ 3 \sin \theta + 4 \cos \theta = 3 \left( \frac{5}{4} \cos \theta \right) + 4 \cos \theta = \frac{15}{4} \cos \theta + 4 \cos \theta. \]

To combine these, we convert \( 4 \cos \theta \) into a fraction:

\[ 4 \cos \theta = \frac{16}{4} \cos \theta. \]

Thus,

\[ 3 \sin \theta + 4 \cos \theta = \left( \frac{15}{4} + \frac{16}{4} \right) \cos \theta = \frac{31}{4} \cos \theta. \]

Now for the denominator:

\[ 3 \sin \theta - 4 \cos \theta = 3 \left( \frac{5}{4} \cos \theta \right) - 4 \cos \theta = \frac{15}{4} \cos \theta - 4 \cos \theta = \frac{15}{4} \cos \theta - \frac{16}{4} \cos \theta = -\frac{1}{4} \cos \theta. \]

### Step 3: Form the complete fraction
Now we can substitute these results back into the fraction:

\[ \frac{3 \sin \theta + 4 \cos \theta}{3 \sin \theta - 4 \cos \theta} = \frac{\frac{31}{4} \cos \theta}{-\frac{1}{4} \cos \theta} = \frac{31}{-1} = -31. \]

### Step 4: Square the result
Now we need to square this result:

\[ \left( -31 \right)^2 = 961. \]

### Final Answer
Thus, the value of

\[ \left( \frac{3 \sin \theta + 4 \cos \theta}{3 \sin \theta - 4 \cos \theta} \right)^2 = 961. \]