If `sin theta - cos theta = 0` then find the value of cot `theta`

Options

A. A.0
B. B.`sqrt""3`
C. C.`1//sqrt""3`
D. D.1

Answer

To solve the equation \( \sin \theta - \cos \theta = 0 \) and find the value of \( \cot \theta \), we can follow these steps:

### Step 1: Set the equation to zero
We start with the given equation:
\[ \sin \theta - \cos \theta = 0 \]

### Step 2: Rearrange the equation
We can rearrange this equation to isolate one of the trigonometric functions:
\[ \sin \theta = \cos \theta \]

### Step 3: Identify the angle
The equation \( \sin \theta = \cos \theta \) is true for specific angles. One of the angles where this holds true is \( \theta = 45^\circ \) (or \( \frac{\pi}{4} \) radians). This is because at \( 45^\circ \), both sine and cosine values are equal:
\[ \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \]

### Step 4: Calculate \( \cot \theta \)
Now, we need to find \( \cot \theta \). The cotangent function is defined as:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Since we have established that \( \theta = 45^\circ \):
\[ \cot 45^\circ = \frac{\cos 45^\circ}{\sin 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \]

### Conclusion
Thus, the value of \( \cot \theta \) is:
\[ \cot \theta = 1 \]