What is the expression `(tan x )/( 1 + sec x) - (tan x)/( 1 - sec x) ` 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 can follow these steps:

### Step 1: Combine the fractions
To combine the two fractions, we need a common denominator. The common denominator for \(1 + \sec x\) and \(1 - \sec x\) is \((1 + \sec x)(1 - \sec x)\).

\[ \frac{\tan x (1 - \sec x) - \tan x (1 + \sec x)}{(1 + \sec x)(1 - \sec x)} \]

### Step 2: Simplify the numerator
Now, simplify 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 3: Simplify the denominator
Now, simplify the denominator:

\[ (1 + \sec x)(1 - \sec x) = 1^2 - (\sec x)^2 = 1 - \sec^2 x \]

### Step 4: Use the identity
Recall the trigonometric identity:

\[ \sec^2 x = 1 + \tan^2 x \]

Thus,

\[ 1 - \sec^2 x = 1 - (1 + \tan^2 x) = -\tan^2 x \]

### Step 5: Substitute back into the expression
Now substitute back into the expression:

\[ \frac{-2 \tan x \sec x}{-\tan^2 x} \]

### Step 6: Cancel out the negatives
The negatives cancel out:

\[ \frac{2 \tan x \sec x}{\tan^2 x} \]

### Step 7: Simplify the expression
Now, simplify the expression:

\[ \frac{2 \sec x}{\tan x} = 2 \cdot \frac{1}{\cos x} \cdot \frac{\cos x}{\sin x} = \frac{2}{\sin x} \]

### Step 8: Final result
Since \(\frac{1}{\sin x} = \csc x\), we have:

\[ \frac{2}{\sin x} = 2 \csc x \]

Thus, the expression \(\frac{\tan x}{1 + \sec x} - \frac{\tan x}{1 - \sec x}\) simplifies to:

\[ \boxed{2 \csc x} \]