`(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 \((1 + \cos x)(1 - \sin x)\). Therefore, we can rewrite the expression as:

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