The simplest value of `sin^(2)x+2tan^(2)x-2sec^(2)x+cos^(2)x` is

Options

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

Answer

To find the simplest value of the expression \( \sin^2 x + 2\tan^2 x - 2\sec^2 x + \cos^2 x \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2 x + \cos^2 x = 1 \]
So, we can substitute \( \sin^2 x + \cos^2 x \) with \( 1 \):
\[ 1 + 2\tan^2 x - 2\sec^2 x \]

### Step 2: Use the Identity for Secant
Recall the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
This can be rearranged to:
\[ \tan^2 x = \sec^2 x - 1 \]
Now, we can substitute \( \tan^2 x \) in our expression:
\[ 1 + 2(\sec^2 x - 1) - 2\sec^2 x \]

### Step 3: Simplify the Expression
Now, let's simplify the expression:
\[ 1 + 2\sec^2 x - 2 - 2\sec^2 x \]
This simplifies to:
\[ 1 - 2 = -1 \]

### Conclusion
Thus, the simplest value of the expression \( \sin^2 x + 2\tan^2 x - 2\sec^2 x + \cos^2 x \) is:
\[ \boxed{-1} \]