Find the value of following.
`tan 142 (1^(@))/(2)`

Answer

To find the value of \( \tan 142.5^\circ \), we can follow these steps:

### Step 1: Rewrite the angle
We can express \( 142.5^\circ \) as:
\[ 142.5^\circ = 90^\circ + 52.5^\circ \]

### Step 2: Use the tangent addition formula
Using the identity \( \tan(90^\circ + \theta) = -\cot(\theta) \), we have:
\[ \tan(142.5^\circ) = \tan(90^\circ + 52.5^\circ) = -\cot(52.5^\circ) \]

### Step 3: Rewrite cotangent
Now, we can express \( \cot(52.5^\circ) \) as:
\[ -\cot(52.5^\circ) = -\frac{1}{\tan(52.5^\circ)} \]

### Step 4: Use the tangent addition formula
Next, we can express \( \tan(52.5^\circ) \) using the angle addition formula:
\[ \tan(52.5^\circ) = \tan(45^\circ + 7.5^\circ) = \frac{\tan(45^\circ) + \tan(7.5^\circ)}{1 - \tan(45^\circ) \tan(7.5^\circ)} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ \tan(52.5^\circ) = \frac{1 + \tan(7.5^\circ)}{1 - \tan(7.5^\circ)} \]

### Step 5: Substitute back
Thus, we can substitute back into our expression for \( \tan(142.5^\circ) \):
\[ \tan(142.5^\circ) = -\frac{1}{\frac{1 + \tan(7.5^\circ)}{1 - \tan(7.5^\circ)}} = -\frac{1 - \tan(7.5^\circ)}{1 + \tan(7.5^\circ)} \]

### Step 6: Rationalize the expression
Now, we can rationalize this expression:
\[ \tan(142.5^\circ) = -\frac{(1 - \tan(7.5^\circ))(1 - \tan(7.5^\circ))}{(1 + \tan(7.5^\circ))(1 - \tan(7.5^\circ))} = -\frac{(1 - \tan(7.5^\circ))^2}{1 - \tan^2(7.5^\circ)} \]

### Step 7: Final expression
Using the identity \( \tan^2(a) + 1 = \sec^2(a) \), we can simplify further, but we can also leave it in this form for practical purposes.

### Conclusion
Thus, the value of \( \tan 142.5^\circ \) is:
\[ \tan(142.5^\circ) = -\frac{(1 - \tan(7.5^\circ))^2}{1 - \tan^2(7.5^\circ)} \]