If `(2sintheta-costheta)/(costheta+sintheta)=1` , then value of `cottheta` is :

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. 3
D. 2

Answer

To solve the equation \(\frac{2\sin\theta - \cos\theta}{\cos\theta + \sin\theta} = 1\) and find the value of \(\cot\theta\), follow these steps:

### Step 1: Cross-Multiply
Start by cross-multiplying the equation to eliminate the fraction:
\[ 2\sin\theta - \cos\theta = \cos\theta + \sin\theta \]

### Step 2: Rearrange the Equation
Now, rearranging the equation gives:
\[ 2\sin\theta - \sin\theta = \cos\theta + \cos\theta \]
This simplifies to:
\[ \sin\theta = 2\cos\theta \]

### Step 3: Express \(\cot\theta\)
Recall that \(\cot\theta = \frac{\cos\theta}{\sin\theta}\). We can substitute \(\sin\theta\) from the previous step:
\[ \cot\theta = \frac{\cos\theta}{2\cos\theta} = \frac{1}{2} \]

### Conclusion
Thus, the value of \(\cot\theta\) is:
\[ \cot\theta = \frac{1}{2} \]

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