Fundamental period of `f(x) = cos^6 x-sin^6 x `is equal to

Options

A. `(2pi)/(3)`
B. `pi`
C. ` (pi)/(3)`
D. `(pi)/(6)`

Answer

To find the fundamental period of the function \( f(x) = \cos^6 x - \sin^6 x \), we can follow these steps:

### Step 1: Understand the periodicity of the components
The functions \( \cos x \) and \( \sin x \) both have a fundamental period of \( 2\pi \). Therefore, any function derived from these will also have a period that is a divisor of \( 2\pi \).

**Hint:** Remember that the period of a function is the smallest positive value \( T \) such that \( f(x + T) = f(x) \) for all \( x \).

### Step 2: Simplify the expression
We can use the identity for the difference of squares to simplify \( f(x) \):
\[ f(x) = \cos^6 x - \sin^6 x = (\cos^2 x - \sin^2 x)(\cos^4 x + \cos^2 x \sin^2 x + \sin^4 x) \]

**Hint:** Look for algebraic identities that can help simplify the expression.

### Step 3: Analyze \( \cos^2 x - \sin^2 x \)
The term \( \cos^2 x - \sin^2 x \) can be rewritten using the double angle identity:
\[ \cos^2 x - \sin^2 x = \cos(2x) \]
This means that \( f(x) \) can be expressed as:
\[ f(x) = \cos(2x)(\cos^4 x + \cos^2 x \sin^2 x + \sin^4 x) \]

**Hint:** Use trigonometric identities to express the function in a simpler form.

### Step 4: Analyze the second term
The term \( \cos^4 x + \cos^2 x \sin^2 x + \sin^4 x \) can be rewritten as:
\[ \cos^4 x + \sin^4 x + \cos^2 x \sin^2 x = (\cos^2 x + \sin^2 x)^2 - \cos^2 x \sin^2 x = 1 - \frac{1}{4} \sin^2(2x) \]
This does not affect the periodicity since it is a constant plus a function of \( \sin(2x) \).

**Hint:** Recognize that \( \sin^2(2x) \) has a period of \( \pi \).

### Step 5: Determine the period of \( f(x) \)
The function \( \cos(2x) \) has a period of \( \pi \). Since the other term \( 1 - \frac{1}{4} \sin^2(2x) \) also has a period of \( \pi \), the overall function \( f(x) \) will have a period of \( \pi \).

**Hint:** The period of a product of functions is the least common multiple of their individual periods.

### Conclusion
Thus, the fundamental period of \( f(x) = \cos^6 x - \sin^6 x \) is \( \pi \).

**Final Answer:** The fundamental period of \( f(x) \) is \( \pi \).