If `sin^2 theta + sin^2 alpha =1`, then what is the value of `cot (theta+alpha)` ?

Options

A. 1
B. `sqrt3`
C. 0
D. `(1)/(sqrt3)`

Answer

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

1. **Given Equation**:
\[ \sin^2 \theta + \sin^2 \alpha = 1 \]

2. **Choosing a Value for Alpha**:
To simplify the problem, we can choose a convenient value for \(\alpha\). Let's set:
\[ \alpha = 0^\circ \]

3. **Substituting Alpha into the Equation**:
Now we substitute \(\alpha = 0^\circ\) into the equation:
\[ \sin^2 \theta + \sin^2 0^\circ = 1 \]
Since \(\sin 0^\circ = 0\), we have:
\[ \sin^2 \theta + 0 = 1 \]
This simplifies to:
\[ \sin^2 \theta = 1 \]

4. **Finding Sine of Theta**:
Taking the square root of both sides, we find:
\[ \sin \theta = 1 \]
This implies:
\[ \theta = 90^\circ \]

5. **Finding Cotangent of (Theta + Alpha)**:
Now we need to find \(\cot(\theta + \alpha)\):
\[ \cot(\theta + \alpha) = \cot(90^\circ + 0^\circ) = \cot(90^\circ) \]

6. **Evaluating Cotangent**:
The cotangent of \(90^\circ\) is:
\[ \cot(90^\circ) = 0 \]

Thus, the final answer is:
\[ \cot(\theta + \alpha) = 0 \]

### Summary of Steps:
1. Start with the equation \(\sin^2 \theta + \sin^2 \alpha = 1\).
2. Choose \(\alpha = 0^\circ\) to simplify the equation.
3. Substitute and simplify to find \(\sin^2 \theta = 1\).
4. Conclude that \(\sin \theta = 1\) leads to \(\theta = 90^\circ\).
5. Calculate \(\cot(90^\circ + 0^\circ)\).
6. Conclude that \(\cot(90^\circ) = 0\).