If `sin(pi cos theta) = cos(pi sin theta)`, then the value of `cos(theta+- pi/4)` is

Options

A. `"cos"(pi)/(4)`
B. `(1)/(2) "cos" (pi)/(4)`
C. `"cos"(pi)/(8)`
D. none of these

Answer

To solve the equation \( \sin(\pi \cos \theta) = \cos(\pi \sin \theta) \) and find the value of \( \cos(\theta \pm \frac{\pi}{4}) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin(\pi \cos \theta) = \cos(\pi \sin \theta) \]

### Step 2: Use the co-function identity
We know that:
\[ \cos(\frac{\pi}{2} - x) = \sin(x) \]
Thus, we can rewrite the right-hand side:
\[ \cos(\pi \sin \theta) = \sin\left(\frac{\pi}{2} - \pi \sin \theta\right) \]
This gives us:
\[ \sin(\pi \cos \theta) = \sin\left(\frac{\pi}{2} - \pi \sin \theta\right) \]

### Step 3: Set the arguments equal
Since the sine function is periodic, we can set the arguments equal to each other:
\[ \pi \cos \theta = \frac{\pi}{2} - \pi \sin \theta + 2k\pi \quad \text{(for some integer } k\text{)} \]
This simplifies to:
\[ \pi \cos \theta + \pi \sin \theta = \frac{\pi}{2} + 2k\pi \]

### Step 4: Divide by \(\pi\)
Dividing the entire equation by \(\pi\):
\[ \cos \theta + \sin \theta = \frac{1}{2} + 2k \]

### Step 5: Express in terms of cosine and sine
We can express \(\cos \theta + \sin \theta\) using the identity:
\[ \cos \theta + \sin \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
Thus, we have:
\[ \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{2} + 2k \]

### Step 6: Solve for \(\sin\)
From this equation, we can isolate \(\sin\):
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}} + \frac{2k}{\sqrt{2}} \]

### Step 7: Find \( \cos(\theta \pm \frac{\pi}{4}) \)
To find \( \cos(\theta \pm \frac{\pi}{4}) \), we use the cosine addition and subtraction formulas:
\[ \cos(\theta \pm \frac{\pi}{4}) = \cos \theta \cos \frac{\pi}{4} \mp \sin \theta \sin \frac{\pi}{4} \]
Since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), we have:
\[ \cos(\theta \pm \frac{\pi}{4}) = \frac{1}{\sqrt{2}} (\cos \theta \mp \sin \theta) \]

### Step 8: Substitute \(\cos \theta + \sin \theta\)
Using our previous result:
\[ \cos \theta + \sin \theta = \frac{1}{2} + 2k \]
We can express \(\cos \theta - \sin \theta\) using the identity:
\[ \cos \theta - \sin \theta = \sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right) \]

### Conclusion
Thus, we conclude that:
\[ \cos(\theta \pm \frac{\pi}{4}) = \frac{1}{\sqrt{2}} \left(\frac{1}{2} + 2k \mp \sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right)\right) \]
The final value of \( \cos(\theta \pm \frac{\pi}{4}) \) is:
\[ \frac{1}{2\sqrt{2}} \]