Period of` f(x) = sin^4 x + cos^4 x`

Options

A. `pi`
B. `pi/2`
C. `2pi`
D. None of these

Answer

To find the period of the function \( f(x) = \sin^4 x + \cos^4 x \), we will follow these steps:

### Step 1: Rewrite the function using trigonometric identities
We know that:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \]
Since \( \sin^2 x + \cos^2 x = 1 \), we can simplify it to:
\[ f(x) = 1 - 2\sin^2 x \cos^2 x \]

### Step 2: Use the double angle identity
We can express \( \sin^2 x \cos^2 x \) using the double angle identity:
\[ \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = 1 - \frac{1}{2} \sin^2(2x) \]

### Step 3: Determine the periodicity of \( f(x) \)
The function \( \sin^2(2x) \) has a period of \( \frac{\pi}{2} \) because the sine function has a period of \( 2\pi \) and the argument \( 2x \) compresses the period by a factor of 2. Therefore, the period of \( \sin^2(2x) \) is:
\[ \frac{2\pi}{2} = \pi \]

### Step 4: Conclude the period of \( f(x) \)
Since \( f(x) \) is a combination of constant and periodic functions, the period of \( f(x) \) is the same as the period of \( \sin^2(2x) \), which is \( \pi \).

Thus, the period of \( f(x) = \sin^4 x + \cos^4 x \) is:
\[ \boxed{\pi} \]