`(sin^(3)x)/(1 + cosx) + (cos^(3)x)/(1 - sinx) =`

Options

A. `sqrt(2)cos((pi)/(4) - x)`
B. `sqrt(2)cos((pi)/(4) + x)`
C. `sqrt(2)sin((pi)/(4) - x)`
D. `sqrt(2)sin((pi)/(4) + x)`

Explanation

To solve the equation

\[ \frac{\sin^3 x}{1 + \cos x} + \frac{\cos^3 x}{1 - \sin x} \]

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((1 + \cos x)(1 - \sin x)\).

### Step 2: Rewrite the fractions
We can rewrite the expression as:

\[ \frac{\sin^3 x (1 - \sin x) + \cos^3 x (1 + \cos x)}{(1 + \cos x)(1 - \sin x)} \]

### Step 3: Expand the numerator
Now, we expand the numerator:

\[ \sin^3 x (1 - \sin x) = \sin^3 x - \sin^4 x \]
\[ \cos^3 x (1 + \cos x) = \cos^3 x + \cos^4 x \]

Combining these gives:

\[ \sin^3 x - \sin^4 x + \cos^3 x + \cos^4 x \]

### Step 4: Rearrange the numerator
We can rearrange the terms in the numerator:

\[ (\sin^3 x + \cos^3 x) + (\cos^4 x - \sin^4 x) \]

### Step 5: Factor the difference of squares
Using the identity \(a^4 - b^4 = (a^2 + b^2)(a - b)(a + b)\), we can factor \(\cos^4 x - \sin^4 x\):

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

Since \(\cos^2 x + \sin^2 x = 1\), we have:

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

### Step 6: Substitute back into the expression
Now substituting back, we have:

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

### Step 7: Use the sum of cubes formula
The sum of cubes can be factored as follows:

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

Since \(\sin^2 x + \cos^2 x = 1\), we can simplify this to:

\[ (\sin x + \cos x)(1 - \sin x \cos x) \]

### Step 8: Combine everything
Now, we combine everything into the numerator:

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

### Step 9: Final expression
We can now write the final expression as:

\[ \frac{(\sin x + \cos x)(1 - \sin x \cos x) + (\cos^2 x - \sin^2 x)}{(1 + \cos x)(1 - \sin x)} \]

### Step 10: Simplify
Finally, we can simplify the entire expression, leading us to the conclusion:

\[ \sin x + \cos x \]

### Final Answer
Thus, the answer to the given expression is:

\[ \sin x + \cos x \]

---