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 Concept:
Convert $\cot$ to $\tan$ using the identity $\cot A = \tan(\frac{\pi}{2} - A)$.

Step 2: Formula Application:
$\tan(\pi \cos \theta) = \tan(\frac{\pi}{2} - \pi \sin \theta)$. $\pi \cos \theta = \frac{\pi}{2} - \pi \sin \theta$.

Step 3: Explanation:
Dividing by $\pi$: $\cos \theta + \sin \theta = \frac{1}{2}$. Divide both sides by $\sqrt{2}$: $\frac{1}{\sqrt{2}}\cos \theta + \frac{1}{\sqrt{2}}\sin \theta = \frac{1}{2\sqrt{2}}$ $\sin(\theta + \frac{\pi}{4}) = \frac{1}{2\sqrt{2}}$.

Step 4: Final Answer:
The value is $\frac{1}{2\sqrt{2}}$.