If `tantheta=sqrt(2)-1`,then find the value of `cottheta`.

Answer

To find the value of \( \cot \theta \) given that \( \tan \theta = \sqrt{2} - 1 \), we can follow these steps:

### Step 1: Use the relationship between cotangent and tangent
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Substituting the given value of \( \tan \theta \):
\[ \cot \theta = \frac{1}{\sqrt{2} - 1} \]

### Step 2: Rationalize the denominator
To simplify \( \cot \theta \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{2} + 1 \):
\[ \cot \theta = \frac{1 \cdot (\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} \]

### Step 3: Simplify the denominator
Using the difference of squares formula:
\[ (\sqrt{2} - 1)(\sqrt{2} + 1) = (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]

### Step 4: Final expression for cotangent
Now substituting back into the equation:
\[ \cot \theta = \frac{\sqrt{2} + 1}{1} = \sqrt{2} + 1 \]

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