What is the value of `[(1)/((1-tantheta))]-[(1)/((1+tantheta))]`?

Options

A. `tantheta`
B. `cot2theta`
C. `tan2theta`
D. `cottheta`

Answer

To find the value of the expression \(\frac{1}{1 - \tan \theta} - \frac{1}{1 + \tan \theta}\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((1 - \tan \theta)(1 + \tan \theta)\).

### Step 2: Rewrite the expression with the common denominator
\[ \frac{1}{1 - \tan \theta} - \frac{1}{1 + \tan \theta} = \frac{(1 + \tan \theta) - (1 - \tan \theta)}{(1 - \tan \theta)(1 + \tan \theta)} \]

### Step 3: Simplify the numerator
Now simplify the numerator:
\[ (1 + \tan \theta) - (1 - \tan \theta) = 1 + \tan \theta - 1 + \tan \theta = 2\tan \theta \]

### Step 4: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 - \tan \theta)(1 + \tan \theta) = 1^2 - (\tan \theta)^2 = 1 - \tan^2 \theta \]

### Step 5: Combine the results
Now we can write the expression as:
\[ \frac{2\tan \theta}{1 - \tan^2 \theta} \]

### Step 6: Recognize the trigonometric identity
Using the double angle identity for tangent, we know that:
\[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]

### Final Result
Thus, we have:
\[ \frac{1}{1 - \tan \theta} - \frac{1}{1 + \tan \theta} = \tan 2\theta \]

### Conclusion
The value of the expression is \(\tan 2\theta\).

---