What is the simplified value of `[ ((sec^(3)x - tan^(3)x ))/((sec x - tan x ))]-2 tan^(2)x - sec x tan x `?

Options

A. 0
B. 2
C. `-1`
D. 1

Answer

To simplify the expression \[ \frac{(\sec^3 x - \tan^3 x)}{(\sec x - \tan x)} - 2 \tan^2 x - \sec x \tan x \], we can follow these steps:

### Step 1: Recognize the formula for the difference of cubes
We know that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, we can let \( a = \sec x \) and \( b = \tan x \).

### Step 2: Apply the difference of cubes formula
Using the formula, we can rewrite the numerator:
\[ \sec^3 x - \tan^3 x = (\sec x - \tan x)(\sec^2 x + \sec x \tan x + \tan^2 x) \]
Thus, we can substitute this into our expression:
\[ \frac{(\sec x - \tan x)(\sec^2 x + \sec x \tan x + \tan^2 x)}{(\sec x - \tan x)} \]

### Step 3: Cancel the common terms
Since \(\sec x - \tan x\) appears in both the numerator and the denominator, we can cancel it out (assuming \(\sec x \neq \tan x\)):
\[ \sec^2 x + \sec x \tan x + \tan^2 x \]

### Step 4: Substitute the identity for \(\sec^2 x\)
We know from trigonometric identities that:
\[ \sec^2 x = 1 + \tan^2 x \]
Substituting this into our expression gives:
\[ (1 + \tan^2 x) + \sec x \tan x + \tan^2 x \]
This simplifies to:
\[ 1 + 2\tan^2 x + \sec x \tan x \]

### Step 5: Combine with the remaining terms
Now, we need to subtract \(2 \tan^2 x + \sec x \tan x\):
\[ (1 + 2\tan^2 x + \sec x \tan x) - (2 \tan^2 x + \sec x \tan x) \]

### Step 6: Simplify the expression
When we subtract, the \(2\tan^2 x\) and \(\sec x \tan x\) terms cancel out:
\[ 1 + 2\tan^2 x + \sec x \tan x - 2\tan^2 x - \sec x \tan x = 1 \]

### Final Answer
Thus, the simplified value of the original expression is:
\[ \boxed{1} \]