If `(costheta+sintheta)/(costheta-sintheta)=8` then the value of cot `theta` is equal to:

Options

A. `7/6`
B. `9/7`
C. `6/5`
D. `8/7`

Answer

To solve the equation \(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = 8\), we can use the method of Component and Dividend (C and D). Here’s a step-by-step solution:

### Step 1: Set up the equation
We start with the given equation:
\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = 8 \]

### Step 2: Cross-multiply
Cross-multiplying gives:
\[ \cos \theta + \sin \theta = 8(\cos \theta - \sin \theta) \]

### Step 3: Expand the right side
Expanding the right side, we have:
\[ \cos \theta + \sin \theta = 8\cos \theta - 8\sin \theta \]

### Step 4: Rearrange the equation
Rearranging the equation to bring all terms involving \(\cos \theta\) and \(\sin \theta\) to one side:
\[ \cos \theta + \sin \theta - 8\cos \theta + 8\sin \theta = 0 \]
This simplifies to:
\[ -7\cos \theta + 9\sin \theta = 0 \]

### Step 5: Isolate \(\cot \theta\)
Rearranging gives:
\[ 7\cos \theta = 9\sin \theta \]
Dividing both sides by \(\sin \theta \cos \theta\) (assuming \(\sin \theta \neq 0\)):
\[ \frac{7\cos \theta}{\sin \theta} = 9 \]
This can be rewritten as:
\[ 7 \cot \theta = 9 \]

### Step 6: Solve for \(\cot \theta\)
Now, divide both sides by 7:
\[ \cot \theta = \frac{9}{7} \]

### Conclusion
Thus, the value of \(\cot \theta\) is:
\[ \cot \theta = \frac{9}{7} \]