If `tan(picostheta)=cot(pisintheta),"then the value of"sin(theta+(pi)/(4))` equals

Options

A. `(1)/(sqrt2)`
B. `(1)/(2sqrt2)`
C. `-(1)/(2sqrt2)`
D. `-(1)/(sqrt2)`

Answer

To solve the problem, we need to find the value of \( \sin\left(\theta + \frac{\pi}{4}\right) \) given that \( \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \]

2. **Rewrite cotangent in terms of tangent:**
\[ \tan(\pi \cos \theta) = \frac{1}{\tan(\pi \sin \theta) \]
This implies:
\[ \tan(\pi \cos \theta) \tan(\pi \sin \theta) = 1 \]

3. **Express tangent in terms of sine and cosine:**
\[ \frac{\sin(\pi \cos \theta)}{\cos(\pi \cos \theta)} \cdot \frac{\sin(\pi \sin \theta)}{\cos(\pi \sin \theta)} = 1 \]

4. **Cross-multiply:**
\[ \sin(\pi \cos \theta) \sin(\pi \sin \theta = \cos(\pi \cos \theta) \cos(\pi \sin \theta) \]

5. **Use the identity for sine and cosine:**
\[ \sin A \sin B = \cos A \cos B \implies \cos(A + B) = 0 \]
Here, \( A = \pi \cos \theta \) and \( B = \pi \sin \theta \).

6. **Set up the equation:**
\[ \pi \cos \theta + \pi \sin \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

7. **Simplify the equation:**
\[ \cos \theta + \sin \theta = \frac{1}{2} \]

8. **Now, we need to find \( \sin\left(\theta + \frac{\pi}{4}\right) \):**
Using the sine addition formula:
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \sin \theta \cos \frac{\pi}{4} + \cos \theta \sin \frac{\pi}{4} \]
Since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \):
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} (\sin \theta + \cos \theta) \]

9. **Substitute the value of \( \sin \theta + \cos \theta \):**
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{1}{2\sqrt{2}} \]

10. **Final answer:**
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}} \]