Expression `(tan x)/(1 + sec x)- (tan x)/(1 -sec x)` is equal to:

Options

A. `"cosec"x`
B. 2 cosec x
C. 2 sin x
D. 2 cos x

Answer

To solve the expression \(\frac{\tan x}{1 + \sec x} - \frac{\tan x}{1 - \sec x}\), we will follow these steps:

### Step 1: Write the expression
The given expression is:
\[ \frac{\tan x}{1 + \sec x} - \frac{\tan x}{1 - \sec x} \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator. The common denominator will be \((1 + \sec x)(1 - \sec x)\).

### Step 3: Rewrite the expression with the common denominator
Now, we can rewrite the expression as:
\[ \frac{\tan x(1 - \sec x) - \tan x(1 + \sec x)}{(1 + \sec x)(1 - \sec x)} \]

### Step 4: Simplify the numerator
Expanding the numerator:
\[ \tan x(1 - \sec x) - \tan x(1 + \sec x) = \tan x - \tan x \sec x - \tan x - \tan x \sec x \]
This simplifies to:
\[ -\tan x \sec x - \tan x \sec x = -2 \tan x \sec x \]

### Step 5: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 + \sec x)(1 - \sec x) = 1^2 - (\sec x)^2 = 1 - \sec^2 x \]
Using the identity \(1 - \sec^2 x = -\tan^2 x\), we can write:
\[ 1 - \sec^2 x = -\tan^2 x \]

### Step 6: Combine the results
Now we can substitute back into the expression:
\[ \frac{-2 \tan x \sec x}{-\tan^2 x} \]
The negatives cancel out, giving us:
\[ \frac{2 \tan x \sec x}{\tan^2 x} \]

### Step 7: Simplify further
This simplifies to:
\[ \frac{2 \sec x}{\tan x} \]
Since \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\), we have:
\[ \frac{2 \cdot \frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{2}{\sin x} = 2 \csc x \]

### Final Answer
Thus, the expression \(\frac{\tan x}{1 + \sec x} - \frac{\tan x}{1 - \sec x}\) is equal to:
\[ \boxed{2 \csc x} \]