If `sintheta_1sintheta_2-costheta_1costheta_2+1=0,` then the value of `tan((theta_1)/2)cot((theta_2)/2)` is equal to `-1` (b) 1 (c) 2 (d) `-2`

Options

A. `a^(2)+b^(2)ge4`
B. `a^(2)+b^(2)le4`
C. `a^(2)+b^(2)ge3`
D. `a^(2)+b^(2)le2`

Explanation

To solve the problem, we start with the given equation:

\[ \sin \theta_1 \sin \theta_2 - \cos \theta_1 \cos \theta_2 + 1 = 0 \]

### Step 1: Rearranging the Equation
We can rearrange the equation as follows:

\[ \sin \theta_1 \sin \theta_2 - \cos \theta_1 \cos \theta_2 = -1 \]

### Step 2: Using the Cosine Addition Formula
Recognizing that the left-hand side can be expressed using the cosine addition formula:

\[ \sin \theta_1 \sin \theta_2 - \cos \theta_1 \cos \theta_2 = -\cos(\theta_1 + \theta_2) \]

Thus, we rewrite the equation:

\[ -\cos(\theta_1 + \theta_2) = -1 \]

### Step 3: Simplifying the Equation
This simplifies to:

\[ \cos(\theta_1 + \theta_2) = 1 \]

### Step 4: Finding the Angles
The cosine function equals 1 at even multiples of \(2\pi\):

\[ \theta_1 + \theta_2 = 2n\pi \quad \text{where } n \text{ is an integer} \]

### Step 5: Finding \(\tan\left(\frac{\theta_1}{2}\right) \cot\left(\frac{\theta_2}{2}\right)\)
We need to find the value of:

\[ \tan\left(\frac{\theta_1}{2}\right) \cot\left(\frac{\theta_2}{2}\right) \]

### Step 6: Substituting \(\theta_2\)
Using the relation from Step 4, we can express \(\theta_2\):

\[ \theta_2 = 2n\pi - \theta_1 \]

Thus, we have:

\[ \tan\left(\frac{\theta_1}{2}\right) \cot\left(\frac{2n\pi - \theta_1}{2}\right) \]

### Step 7: Simplifying \(\cot\left(\frac{2n\pi - \theta_1}{2}\right)\)
Using the cotangent identity:

\[ \cot\left(\frac{2n\pi - \theta_1}{2}\right) = \cot\left(n\pi - \frac{\theta_1}{2}\right) = -\cot\left(\frac{\theta_1}{2}\right) \]

### Step 8: Final Calculation
Now substituting this back into our expression:

\[ \tan\left(\frac{\theta_1}{2}\right) \cdot (-\cot\left(\frac{\theta_1}{2}\right)) \]

Since \(\tan x \cdot \cot x = 1\):

\[ = -1 \]

### Conclusion
Thus, the value of \(\tan\left(\frac{\theta_1}{2}\right) \cot\left(\frac{\theta_2}{2}\right)\) is:

\[ \boxed{-1} \]