If `f(x)=(sin^(4)x+cos^(2)x)/(sin^(2)x+cos^(4)x)"for "x in R`, then f(2010)

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( f(2010) \) for the function

\[ f(x) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}, \]

we will simplify the expression step by step.

### Step 1: Rewrite the function
We start with the function as given:

\[ f(x) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}. \]

### Step 2: Substitute \(\cos^2 x\) and \(\sin^2 x\)
Recall that \(\sin^2 x + \cos^2 x = 1\). We can express \(\cos^2 x\) as \(1 - \sin^2 x\) and \(\sin^2 x\) as \(1 - \cos^2 x\).

### Step 3: Simplify the numerator
The numerator can be rewritten as:

\[ \sin^4 x + \cos^2 x = \sin^4 x + (1 - \sin^2 x) = \sin^4 x + 1 - \sin^2 x. \]

### Step 4: Simplify the denominator
The denominator can be rewritten as:

\[ \sin^2 x + \cos^4 x = \sin^2 x + (1 - \sin^2 x)^2. \]

Expanding \((1 - \sin^2 x)^2\):

\[ (1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x. \]

Thus, the denominator becomes:

\[ \sin^2 x + 1 - 2\sin^2 x + \sin^4 x = 1 - \sin^2 x + \sin^4 x. \]

### Step 5: Combine and simplify
Now we have:

\[ f(x) = \frac{\sin^4 x + 1 - \sin^2 x}{1 - \sin^2 x + \sin^4 x}. \]

Notice that both the numerator and denominator have the same terms. Therefore, we can simplify:

\[ f(x) = \frac{1}{1} = 1. \]

### Step 6: Conclusion
Since \( f(x) = 1 \) for all \( x \in \mathbb{R} \), we find that:

\[ f(2010) = 1. \]

Thus, the final answer is:

\[ \boxed{1}. \]