If `sin (picot theta)=cos (pi tantheta),` then

Options

A. `cot2theta=1/4,3/4`
B. `cot2theta=4, 4/3`
C. `cot2theta=-3/4,-1/4`
D. none of these

Answer

To solve the equation \( \sin(\pi \cot \theta) = \cos(\pi \tan \theta) \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \). Therefore, we can rewrite the equation as:
\[ \sin(\pi \cot \theta) = \sin\left(\frac{\pi}{2} - \pi \tan \theta\right) \]

### Step 2: Set the arguments of sine equal to each other
Since the sine function is periodic, we can set the arguments equal to each other:
\[ \pi \cot \theta = \frac{\pi}{2} - \pi \tan \theta + 2k\pi \quad \text{(for any integer } k\text{)} \]
or
\[ \pi \cot \theta = \pi \tan \theta + \frac{\pi}{2} + 2k\pi \]

### Step 3: Simplify the first case
Taking the first case:
\[ \pi \cot \theta + \pi \tan \theta = \frac{\pi}{2} + 2k\pi \]
Dividing through by \( \pi \):
\[ \cot \theta + \tan \theta = \frac{1}{2} + 2k \]

### Step 4: Use the identity for cotangent and tangent
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \):
\[ \frac{1}{\tan \theta} + \tan \theta = \frac{1}{2} + 2k \]
Let \( x = \tan \theta \):
\[ \frac{1}{x} + x = \frac{1}{2} + 2k \]
Multiplying through by \( x \):
\[ 1 + x^2 = \left(\frac{1}{2} + 2k\right)x \]
Rearranging gives:
\[ x^2 - \left(\frac{1}{2} + 2k\right)x + 1 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{\left(\frac{1}{2} + 2k\right) \pm \sqrt{\left(\frac{1}{2} + 2k\right)^2 - 4}}{2} \]

### Step 6: Find \( \cot(2\theta) \)
Using the double angle formula:
\[ \tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \]
Thus,
\[ \cot(2\theta) = \frac{1 - \tan^2 \theta}{2\tan \theta} \]
Substituting \( x = \tan \theta \):
\[ \cot(2\theta) = \frac{1 - x^2}{2x} \]

### Step 7: Evaluate for specific values of \( k \)
For \( k = 0 \):
1. Solve \( x^2 - \frac{1}{2}x + 1 = 0 \)
2. Calculate \( \cot(2\theta) \) using the values of \( x \).

### Step 8: Check for other values of \( k \)
Repeat the process for \( k = 1 \) and \( k = -1 \) to find other possible values of \( \cot(2\theta) \).

### Conclusion
After evaluating the possible values of \( \cot(2\theta) \), we find that the values are \( \frac{1}{4} \) and \( \frac{3}{4} \).