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 can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that \( \cot x = \tan\left(\frac{\pi}{2} - x\right) \). Therefore, we can rewrite the right-hand side of the equation:
\[ \cot\left(\frac{\pi}{2} \cos \theta\right) = \tan\left(\frac{\pi}{2} - \frac{\pi}{2} \cos \theta\right) = \tan\left(\frac{\pi}{2}(1 - \cos \theta)\right) \]
Now, our equation becomes:
\[ \tan\left(\frac{\pi}{2} \sin \theta\right) = \tan\left(\frac{\pi}{2}(1 - \cos \theta)\right) \]

### Step 2: Set the arguments equal to each other
Since the tangent function is periodic, we can set the arguments equal to each other:
\[ \frac{\pi}{2} \sin \theta = \frac{\pi}{2}(1 - \cos \theta) + m\pi \]
where \( m \) is any integer.

### Step 3: Simplify the equation
Dividing through by \( \frac{\pi}{2} \):
\[ \sin \theta = 1 - \cos \theta + 2m \]
Rearranging gives:
\[ \sin \theta + \cos \theta = 2m + 1 \]

### Step 4: Analyze possible values of \( \sin \theta + \cos \theta \)
The expression \( \sin \theta + \cos \theta \) can be rewritten using the identity:
\[ \sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
This means that the maximum value of \( \sin \theta + \cos \theta \) is \( \sqrt{2} \) and the minimum value is \( -\sqrt{2} \).

### Step 5: Consider integer values for \( m \)
From \( \sin \theta + \cos \theta = 2m + 1 \), we can find possible values for \( m \):
- If \( m = 0 \), then \( \sin \theta + \cos \theta = 1 \).
- If \( m = -1 \), then \( \sin \theta + \cos \theta = -1 \).
- If \( m = 1 \), then \( \sin \theta + \cos \theta = 3 \) (not possible since it exceeds \( \sqrt{2} \)).
- If \( m = -2 \), then \( \sin \theta + \cos \theta = -3 \) (not possible since it is less than \( -\sqrt{2} \)).

### Conclusion
Thus, the possible values for \( \sin \theta + \cos \theta \) are:
\[ \sin \theta + \cos \theta = 1 \quad \text{or} \quad \sin \theta + \cos \theta = -1 \]

### Final Answer
The values of \( \sin \theta + \cos \theta \) are \( 1 \) or \( -1 \).

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