The expression
`(tan(x-(pi)/(2)).cos((3pi)/(2)+x)-sin^(3)((7pi)/(2)-x))/(cos(x-(pi)/(2)).tan((3pi)/(2)+x))`simplifies to

Options

A. `(1+cos^(2)x)`
B. `sin^(2)x`
C. `-(1+cos^(2)x)`
D. `cos^(2)x`

Explanation

To simplify the expression

\[ \frac{\tan\left(x - \frac{\pi}{2}\right) \cos\left(\frac{3\pi}{2} + x\right) - \sin^3\left(\frac{7\pi}{2} - x\right)}{\cos\left(x - \frac{\pi}{2}\right) \tan\left(\frac{3\pi}{2} + x\right)}, \]

we will follow these steps:

### Step 1: Simplify \(\tan\left(x - \frac{\pi}{2}\right)\)

Using the identity \(\tan\left(\theta - \frac{\pi}{2}\right) = -\cot(\theta)\), we have:

\[ \tan\left(x - \frac{\pi}{2}\right) = -\cot(x). \]

### Step 2: Simplify \(\cos\left(\frac{3\pi}{2} + x\right)\)

Using the identity \(\cos\left(\frac{3\pi}{2} + x\right) = -\sin(x)\), we get:

\[ \cos\left(\frac{3\pi}{2} + x\right) = -\sin(x). \]

### Step 3: Substitute into the expression

Now substituting these results into the expression, we have:

\[ \frac{-\cot(x)(-\sin(x)) - \sin^3\left(\frac{7\pi}{2} - x\right)}{\cos\left(x - \frac{\pi}{2}\right) \tan\left(\frac{3\pi}{2} + x\right)}. \]

### Step 4: Simplify \(\sin^3\left(\frac{7\pi}{2} - x\right)\)

Using the identity \(\sin\left(\frac{7\pi}{2} - x\right) = \sin\left(3\pi - x\right) = -\sin(x)\), we find:

\[ \sin^3\left(\frac{7\pi}{2} - x\right) = (-\sin(x))^3 = -\sin^3(x). \]

### Step 5: Substitute \(\cos\left(x - \frac{\pi}{2}\right)\)

Using the identity \(\cos\left(x - \frac{\pi}{2}\right) = \sin(x)\), we have:

\[ \cos\left(x - \frac{\pi}{2}\right) = \sin(x). \]

### Step 6: Simplify \(\tan\left(\frac{3\pi}{2} + x\right)\)

Using the identity \(\tan\left(\frac{3\pi}{2} + x\right) = -\cot(x)\), we find:

\[ \tan\left(\frac{3\pi}{2} + x\right) = -\cot(x). \]

### Step 7: Substitute into the expression

Now substituting these results back into the expression gives:

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

### Step 8: Simplify the numerator

The numerator simplifies to:

\[ \cot(x)\sin(x) + \sin^3(x) = \sin(x)(\cot(x) + \sin^2(x)). \]

### Step 9: Substitute and simplify the entire expression

Now substituting this into the expression gives:

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

### Step 10: Cancel \(\sin(x)\)

Assuming \(\sin(x) \neq 0\), we can cancel \(\sin(x)\):

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

### Step 11: Final simplification

Using \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), we can rewrite the expression:

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

Thus, the expression simplifies to:

\[ \sin^2(x). \]

### Final Result

The simplified expression is:

\[ \sin^2(x). \]