If `1+tantheta=sqrt(2)` , then value of `cot theta-1` is

Options

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

Answer

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

**Step 1: Start with the equation**
\[ 1 + \tan \theta = \sqrt{2} \]

**Step 2: Isolate \(\tan \theta\)**
Subtract 1 from both sides:
\[ \tan \theta = \sqrt{2} - 1 \]

**Step 3: Find \(\cot \theta\)**
Recall that \(\cot \theta\) is the reciprocal of \(\tan \theta\):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{2} - 1} \]

**Step 4: Rationalize the denominator**
To simplify \(\cot \theta\), multiply the numerator and denominator by the conjugate of the denominator:
\[ \cot \theta = \frac{1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{(\sqrt{2})^2 - (1)^2} \]

**Step 5: Simplify the denominator**
Calculate the denominator:
\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]

**Step 6: Final expression for \(\cot \theta\)**
Thus, we have:
\[ \cot \theta = \sqrt{2} + 1 \]

**Step 7: Find \(\cot \theta - 1\)**
Now, we need to find \(\cot \theta - 1\):
\[ \cot \theta - 1 = (\sqrt{2} + 1) - 1 = \sqrt{2} \]

**Final Answer:**
The value of \(\cot \theta - 1\) is:
\[ \sqrt{2} \]

---