If `cos A=1/2 and sin B =1/sqrt2` , find the value of : `(tanA-tanB)/(1+tanAtanB)` : Here angles A and B from different right triangle .

Answer

To solve the problem, we need to find the value of the expression \((\tan A - \tan B) / (1 + \tan A \tan B)\) given that \(\cos A = \frac{1}{2}\) and \(\sin B = \frac{1}{\sqrt{2}}\).

### Step-by-Step Solution:

1. **Find \(\tan A\)**:
- We know that \(\tan A = \frac{\sin A}{\cos A}\).
- First, we need to find \(\sin A\) using the identity \(\sin^2 A + \cos^2 A = 1\).
- Given \(\cos A = \frac{1}{2}\):
\[ \sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \]
\[ \sin A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
- Now, calculate \(\tan A\):
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

2. **Find \(\tan B\)**:
- We know that \(\tan B = \frac{\sin B}{\cos B}\).
- First, we need to find \(\cos B\) using the identity \(\sin^2 B + \cos^2 B = 1\).
- Given \(\sin B = \frac{1}{\sqrt{2}}\):
\[ \cos^2 B = 1 - \sin^2 B = 1 - \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2} \]
\[ \cos B = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]
- Now, calculate \(\tan B\):
\[ \tan B = \frac{\sin B}{\cos B} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \]

3. **Substitute \(\tan A\) and \(\tan B\) into the expression**:
- We need to find:
\[ \frac{\tan A - \tan B}{1 + \tan A \tan B} = \frac{\sqrt{3} - 1}{1 + \sqrt{3} \cdot 1} \]
\[ = \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \]

4. **Rationalize the denominator**:
- Multiply the numerator and the denominator by the conjugate of the denominator:
\[ = \frac{(\sqrt{3} - 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} \]
- Calculate the numerator:
\[ = (\sqrt{3} - 1)(1 - \sqrt{3}) = \sqrt{3} - 3 - 1 + \sqrt{3} = 2\sqrt{3} - 4 \]
- Calculate the denominator:
\[ = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \]
- Therefore, we have:
\[ = \frac{2\sqrt{3} - 4}{-2} = -\left(\sqrt{3} - 2\right) = 2 - \sqrt{3} \]

### Final Answer:
The value of \(\frac{\tan A - \tan B}{1 + \tan A \tan B}\) is \(2 - \sqrt{3}\).