If `(sin theta + cos theta)/(sin theta - cos theta) = (5)/(4)` , the value of `(tan^(2) theta + 1)/(tan^(2) theta - 1) ` is

Options

A. `(25)/(16)`
B. `(41)/(9)`
C. `(41)/(40)`
D. `(40)/(41)`

Answer

To solve the equation \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{5}{4}\) and find the value of \(\frac{\tan^2 \theta + 1}{\tan^2 \theta - 1}\), we can follow these steps:

### Step 1: Cross-multiply the given equation
Starting from the equation:
\[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{5}{4} \]
Cross-multiplying gives:
\[ 4(\sin \theta + \cos \theta) = 5(\sin \theta - \cos \theta) \]

### Step 2: Expand both sides
Expanding both sides results in:
\[ 4\sin \theta + 4\cos \theta = 5\sin \theta - 5\cos \theta \]

### Step 3: Rearrange the equation
Rearranging the equation to isolate terms involving \(\sin \theta\) and \(\cos \theta\):
\[ 4\sin \theta + 4\cos \theta - 5\sin \theta + 5\cos \theta = 0 \]
This simplifies to:
\[ - \sin \theta + 9\cos \theta = 0 \]

### Step 4: Solve for \(\tan \theta\)
From the equation \(-\sin \theta + 9\cos \theta = 0\), we can express \(\tan \theta\):
\[ \sin \theta = 9\cos \theta \implies \tan \theta = \frac{\sin \theta}{\cos \theta} = 9 \]

### Step 5: Calculate \(\tan^2 \theta\)
Now, we find \(\tan^2 \theta\):
\[ \tan^2 \theta = 9^2 = 81 \]

### Step 6: Substitute into the expression
Now we substitute \(\tan^2 \theta\) into the expression \(\frac{\tan^2 \theta + 1}{\tan^2 \theta - 1}\):
\[ \frac{\tan^2 \theta + 1}{\tan^2 \theta - 1} = \frac{81 + 1}{81 - 1} = \frac{82}{80} \]

### Step 7: Simplify the fraction
Finally, simplifying \(\frac{82}{80}\):
\[ \frac{82}{80} = \frac{41}{40} \]

Thus, the value of \(\frac{\tan^2 \theta + 1}{\tan^2 \theta - 1}\) is \(\frac{41}{40}\).

### Final Answer:
\[ \frac{41}{40} \]