Question

If $\tan(\pi \cos \theta) = \cot(\pi \sin \theta)$, then $\sin\left(\frac{\pi}{4} + \theta\right) = $ ______.

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{4}$
• D. $\frac{1}{2\sqrt{2}}$

Hint:

Multiply any expression of the form $\sin \theta + \cos \theta = X$ by $\frac{1}{\sqrt{2}}$ on both sides, and it will instantly collapse into the identity $\sin(\theta + \frac{\pi}{4}) = \frac{X}{\sqrt{2}}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a trigonometric equality involving tangent and cotangent. We must manipulate this equation to find the value of a specific compound angle sine function.

Step 2: Detailed Explanation:
The given equation is:
$\tan(\pi \cos \theta) = \cot(\pi \sin \theta)$
Using the complementary angle identity $\cot(X) = \tan\left(\frac{\pi}{2} - X\right)$, rewrite the right side:
$\tan(\pi \cos \theta) = \tan\left(\frac{\pi}{2} - \pi \sin \theta\right)$
Equating the principal angles (for the simplest general solution where $n=0$):
$\pi \cos \theta = \frac{\pi}{2} - \pi \sin \theta$
Divide the entire equation by $\pi$:
$\cos \theta = \frac{1}{2} - \sin \theta$
Rearrange to group the trigonometric terms:
$\sin \theta + \cos \theta = \frac{1}{2}$ --- (Equation 1)
The target expression we need to evaluate is:
$E = \sin\left(\frac{\pi}{4} + \theta\right)$
Apply the sine addition formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$:
$E = \sin\left(\frac{\pi}{4}\right) \cos \theta + \cos\left(\frac{\pi}{4}\right) \sin \theta$
Since $\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$:
$E = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta$
$E = \frac{1}{\sqrt{2}} (\sin \theta + \cos \theta)$
Now, substitute the value from Equation 1 ($\sin \theta + \cos \theta = \frac{1}{2}$):
$E = \frac{1}{\sqrt{2}} \left( \frac{1}{2} \right)$
$E = \frac{1}{2\sqrt{2}}$

Step 3: Final Answer:
The value is $\frac{1}{2\sqrt{2}}$, matching option (d).