Question

If \( f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x} \) for \( x \in \mathbb{R} \), then find \( f(2023) \).

• A. 1
• B. 0
• C. 2
• D. \( \pi \)

Hint:

When dealing with trigonometric functions in expressions, always look for periodicity and symmetry to simplify the problem.

Answer 1

The Correct Option is A

The given function is:
\[ f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x} \] We need to find the value of \( f(2023) \). Let's begin by simplifying the function. Notice that \( f(x) \) contains trigonometric terms. We will attempt to evaluate it at \( x = 2023 \). The key observation here is that the numerator and denominator share similar forms, and since trigonometric functions like sine and cosine repeat periodically, the values of \( \cos^2 x \) and \( \sin^2 x \) will follow the same periodic pattern. For \( x = 2023 \), since the periodic nature of sine and cosine ensures that both \( \cos^2 x \) and \( \sin^2 x \) have values that result in the overall simplification of the expression, we can evaluate the expression as: \[ f(2023) = 1 \] Thus, the value of \( f(2023) \) is \( 1 \).