If `sin[(pi)/(4)cottheta] = cos[(pi)/(4)tantheta]` then `theta` can be :

Options

A. `npi+(pi)/(2)`
B. `npi +(pi)/(4)`
C. `npi -(pi)/(4)`
D. `npi +(pi)/(3)`

Answer

To solve the equation \( \sin\left(\frac{\pi}{4} \cot \theta\right) = \cos\left(\frac{\pi}{4} \tan \theta\right) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin\left(\frac{\pi}{4} \cot \theta\right) = \cos\left(\frac{\pi}{4} \tan \theta\right) \]

### Step 2: Use the identity for cosine
Using the identity \( \cos x = \sin\left(\frac{\pi}{2} - x\right) \), we can rewrite the right-hand side:
\[ \sin\left(\frac{\pi}{4} \cot \theta\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{4} \tan \theta\right) \]

### Step 3: Set the angles equal
Since the sine function is equal, we can set the angles equal to each other:
\[ \frac{\pi}{4} \cot \theta = \frac{\pi}{2} - \frac{\pi}{4} \tan \theta \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ \frac{\pi}{4} \cot \theta + \frac{\pi}{4} \tan \theta = \frac{\pi}{2} \]
Factoring out \( \frac{\pi}{4} \):
\[ \frac{\pi}{4} \left(\cot \theta + \tan \theta\right) = \frac{\pi}{2} \]

### Step 5: Simplify the equation
Dividing both sides by \( \frac{\pi}{4} \):
\[ \cot \theta + \tan \theta = 2 \]

### Step 6: Substitute cotangent and tangent
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = 2 \]

### Step 7: Combine the fractions
Combining the fractions gives:
\[ \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = 2 \]
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \frac{1}{\sin \theta \cos \theta} = 2 \]

### Step 8: Cross-multiply
Cross-multiplying gives:
\[ 1 = 2 \sin \theta \cos \theta \]

### Step 9: Use the double angle identity
Using the double angle identity \( 2 \sin \theta \cos \theta = \sin(2\theta) \):
\[ \sin(2\theta) = 1 \]

### Step 10: Solve for \( 2\theta \)
The general solution for \( \sin x = 1 \) is:
\[ 2\theta = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]

### Step 11: Solve for \( \theta \)
Dividing by 2 gives:
\[ \theta = \frac{\pi}{4} + n\pi \]

### Final Answer
Thus, the values of \( \theta \) can be expressed as:
\[ \theta = n\pi + \frac{\pi}{4} \quad (n \in \mathbb{Z}) \]