Let `f_k(x) = 1/k(sin^k x + cos^k x)` where `x in RR` and `k gt= 1.` Then `f_4(x) - f_6(x)` equals

Answer

To solve the problem, we need to find the expression for \( f_4(x) - f_6(x) \) where \( f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x) \).

### Step 1: Write the expressions for \( f_4(x) \) and \( f_6(x) \)

\[ f_4(x) = \frac{1}{4} (\sin^4 x + \cos^4 x) \]
\[ f_6(x) = \frac{1}{6} (\sin^6 x + \cos^6 x) \]

### Step 2: Simplify \( f_4(x) \)

Using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):

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

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

Thus,

\[ f_4(x) = \frac{1}{4} (1 - 2\sin^2 x \cos^2 x) \]

### Step 3: Simplify \( f_6(x) \)

Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):

\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]
Again, since \( \sin^2 x + \cos^2 x = 1 \):

\[ \sin^6 x + \cos^6 x = 1(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]
Substituting \( \sin^4 x + \cos^4 x \):

\[ \sin^6 x + \cos^6 x = \sin^4 x + \cos^4 x - \sin^2 x \cos^2 x \]
Using the previous result for \( \sin^4 x + \cos^4 x \):

\[ \sin^6 x + \cos^6 x = (1 - 2\sin^2 x \cos^2 x) - \sin^2 x \cos^2 x = 1 - 3\sin^2 x \cos^2 x \]

Thus,

\[ f_6(x) = \frac{1}{6} (1 - 3\sin^2 x \cos^2 x) \]

### Step 4: Calculate \( f_4(x) - f_6(x) \)

Now, we can find \( f_4(x) - f_6(x) \):

\[ f_4(x) - f_6(x) = \left(\frac{1}{4}(1 - 2\sin^2 x \cos^2 x)\right) - \left(\frac{1}{6}(1 - 3\sin^2 x \cos^2 x)\right) \]

### Step 5: Find a common denominator and simplify

The common denominator for \( 4 \) and \( 6 \) is \( 12 \):

\[ f_4(x) - f_6(x) = \frac{3(1 - 2\sin^2 x \cos^2 x)}{12} - \frac{2(1 - 3\sin^2 x \cos^2 x)}{12} \]

Combining the fractions:

\[ = \frac{3(1 - 2\sin^2 x \cos^2 x) - 2(1 - 3\sin^2 x \cos^2 x)}{12} \]

Distributing:

\[ = \frac{3 - 6\sin^2 x \cos^2 x - 2 + 6\sin^2 x \cos^2 x}{12} \]

Combining like terms:

\[ = \frac{1}{12} \]

### Final Answer

Thus, the final result is:

\[ f_4(x) - f_6(x) = \frac{1}{12} \]

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