Question

If $ \sin(\pi \cos \theta) = \cos(\pi \sin \theta) $, then the value of $ \cos \left( \theta + \frac{\pi}{4} \right) $

• A. \( \frac{1}{\sqrt{2}} \)
• B. \( \frac{2}{\sqrt{2}} \)
• C. \( \frac{1}{\sqrt{2}} \)
• D. \( -\frac{1}{\sqrt{2}} \)

Hint:

In trigonometric equations, symmetry and specific angle values like \( \frac{\pi}{4} \) can often simplify the equation and lead to a solution.

Answer 1

The Correct Option is A

We are given the equation: \[ \sin(\pi \cos \theta) = \cos(\pi \sin \theta) \]
Step 1: Use of Symmetry
The equation \( \sin(\pi \cos \theta) = \cos(\pi \sin \theta) \) suggests that the values of \( \pi \cos \theta \) and \( \pi \sin \theta \) are related in such a way that symmetry can be applied.
Specifically, this indicates that \( \cos \theta \) and \( \sin \theta \) may have specific values at certain points for which the equation holds true.
For instance, \( \theta = \frac{\pi}{4} \).
Step 2: Evaluate Trigonometric Identity
For \( \theta = \frac{\pi}{4} \), we know: \[ \sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \] This simplifies the equation, and both sides match.
Hence, for \( \theta = \frac{\pi}{4} \), the value of \( \cos \left( \theta + \frac{\pi}{4} \right) \) becomes: \[ \cos \left( \theta + \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \]
Step 3: Conclusion
Thus, the correct answer is \( \frac{1}{\sqrt{2}} \), which corresponds to option (a).