If `tantheta-cottheta=0` , find the value of `sintheta+costheta`.

Options

A. 0
B. 1
C. `sqrt(2)`
D. 2

Answer

To solve the equation \( \tan \theta - \cot \theta = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan \theta - \cot \theta = 0 \]
This implies:
\[ \tan \theta = \cot \theta \]

### Step 2: Use the definition of tangent and cotangent
Recall that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta}{\sin \theta} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ \sin^2 \theta = \cos^2 \theta \]

### Step 4: Use the Pythagorean identity
From the equation \( \sin^2 \theta = \cos^2 \theta \), we can express this as:
\[ \sin^2 \theta - \cos^2 \theta = 0 \]
This can be factored as:
\[ (\sin \theta - \cos \theta)(\sin \theta + \cos \theta) = 0 \]

### Step 5: Solve for \( \theta \)
This gives us two cases:
1. \( \sin \theta - \cos \theta = 0 \) which implies \( \sin \theta = \cos \theta \)
2. \( \sin \theta + \cos \theta = 0 \) which implies \( \sin \theta = -\cos \theta \)

For the first case \( \sin \theta = \cos \theta \), we know this occurs at:
\[ \theta = 45^\circ \quad \text{or} \quad \theta = 225^\circ \]

For the second case \( \sin \theta = -\cos \theta \), this occurs at:
\[ \theta = 135^\circ \quad \text{or} \quad \theta = 315^\circ \]

### Step 6: Find \( \sin \theta + \cos \theta \)
Now we calculate \( \sin \theta + \cos \theta \) for the angles found:

1. For \( \theta = 45^\circ \):
\[ \sin 45^\circ + \cos 45^\circ = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \]

2. For \( \theta = 135^\circ \):
\[ \sin 135^\circ + \cos 135^\circ = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0 \]

3. For \( \theta = 225^\circ \):
\[ \sin 225^\circ + \cos 225^\circ = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2} \]

4. For \( \theta = 315^\circ \):
\[ \sin 315^\circ + \cos 315^\circ = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0 \]

### Final Answer
Thus, the values of \( \sin \theta + \cos \theta \) can be \( \sqrt{2}, 0, -\sqrt{2} \).