`(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. None of these

Answer

To solve the equation

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

we will simplify each term step by step.

### Step 1: Rewrite the terms
We start with the expression:

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

### Step 2: Factor the cubes
We can use the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) to factor the terms. Here, we can express \(\sin^3 x\) and \(\cos^3 x\) in a useful form.

### Step 3: Combine the fractions
To combine these fractions, we need a common denominator. The common denominator will be \((1 + \cos x)(1 - \sin x)\). Thus, we rewrite the expression as:

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

### Step 4: 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 us:

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

### Step 5: Use the Pythagorean identity
Recall that \(\sin^2 x + \cos^2 x = 1\). We can express \(\sin^4 x\) and \(\cos^4 x\) in terms of \(\sin^2 x\) and \(\cos^2 x\):

\[ \sin^4 x = (\sin^2 x)^2, \]
\[ \cos^4 x = (\cos^2 x)^2. \]

### Step 6: Combine like terms
We can now combine the terms:

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

### Step 7: Factor the cubes and squares
Using the identity for cubes and the difference of squares:

\[ \sin^3 x + \cos^3 x = (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x), \]
and
\[ \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = (\cos^2 x - \sin^2 x)(1). \]

### Step 8: Substitute back into the expression
Now we substitute back into our expression:

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

### Step 9: Final simplification
We can simplify further if needed, but this gives us a clear expression.

Thus, the final answer is:

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