`lim_(xtooo) (sin^(4)x-sin^(2)x+1)/(cos^(4)x-cos^(2)x+1)`is equal to

Options

A. 0
B. 1
C. 10
D. 100

Explanation

To solve the limit \( \lim_{x \to \infty} \frac{\sin^4 x - \sin^2 x + 1}{\cos^4 x - \cos^2 x + 1} \), we will follow these steps:

### Step 1: Rewrite the limit
Let \( L = \lim_{x \to \infty} \frac{\sin^4 x - \sin^2 x + 1}{\cos^4 x - \cos^2 x + 1} \).

### Step 2: Use the identity \( \sin^2 x + \cos^2 x = 1 \)
Since \( \sin^2 x + \cos^2 x = 1 \), we can express \( \sin^2 x \) in terms of \( \cos^2 x \):
\[ \sin^2 x = 1 - \cos^2 x. \]

### Step 3: Substitute \( \sin^2 x \) in the numerator
Now, substitute \( \sin^2 x \) in the numerator:
\[ \sin^4 x = (1 - \cos^2 x)^2 = 1 - 2\cos^2 x + \cos^4 x. \]
Thus, the numerator becomes:
\[ \sin^4 x - \sin^2 x + 1 = (1 - 2\cos^2 x + \cos^4 x) - (1 - \cos^2 x) + 1. \]
Simplifying this:
\[ = 1 - 2\cos^2 x + \cos^4 x - 1 + \cos^2 x + 1 = \cos^4 x - \cos^2 x + 1. \]

### Step 4: Rewrite the limit with the new numerator
Now, we can rewrite the limit:
\[ L = \lim_{x \to \infty} \frac{\cos^4 x - \cos^2 x + 1}{\cos^4 x - \cos^2 x + 1}. \]

### Step 5: Simplify the expression
Since the numerator and denominator are identical, we have:
\[ L = \lim_{x \to \infty} 1 = 1. \]

### Final Answer
Thus, the limit is:
\[ \boxed{1}. \]