If ` tan((pi)/(2) sin theta )= cot((pi)/(2) cos theta )`, then ` sin theta + cos theta ` is equal to

Options

A. 0
B. 1
C. `-1`
D. 1 or -1

Answer

To solve the equation \( \tan\left(\frac{\pi}{2} \sin \theta\right) = \cot\left(\frac{\pi}{2} \cos \theta\right) \), we will follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
Using the identity \( \cot x = \tan\left(\frac{\pi}{2} - x\right) \), we can rewrite the right side of the equation:
\[ \cot\left(\frac{\pi}{2} \cos \theta\right) = \tan\left(\frac{\pi}{2} - \frac{\pi}{2} \cos \theta\right) = \tan\left(\frac{\pi}{2}(1 - \cos \theta)\right) \]
Thus, our equation becomes:
\[ \tan\left(\frac{\pi}{2} \sin \theta\right) = \tan\left(\frac{\pi}{2}(1 - \cos \theta)\right) \]

### Step 2: Set the angles equal
Since the tangent function is periodic, we can equate the angles:
\[ \frac{\pi}{2} \sin \theta = \frac{\pi}{2}(1 - \cos \theta) + n\pi \quad (n \in \mathbb{Z}) \]
For simplicity, we will consider the principal values first (i.e., \( n = 0 \)):
\[ \frac{\pi}{2} \sin \theta = \frac{\pi}{2}(1 - \cos \theta) \]

### Step 3: Simplify the equation
Dividing both sides by \( \frac{\pi}{2} \) (assuming \( \pi \neq 0 \)):
\[ \sin \theta = 1 - \cos \theta \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ \sin \theta + \cos \theta = 1 \]

### Step 5: Consider the second case
Now let's consider the case when \( n = 1 \):
\[ \frac{\pi}{2} \sin \theta = \frac{\pi}{2}(1 - \cos \theta) + \pi \]
This simplifies to:
\[ \sin \theta = 1 - \cos \theta + 2 \]
Thus,
\[ \sin \theta + \cos \theta = -1 \]

### Conclusion
The possible values for \( \sin \theta + \cos \theta \) are:
1. \( \sin \theta + \cos \theta = 1 \)
2. \( \sin \theta + \cos \theta = -1 \)

Thus, the answer is that \( \sin \theta + \cos \theta \) can be either \( 1 \) or \( -1 \).