`(sin(270^(@)+theta).cos(360^(@)+theta).tan(180^(@)+theta))/(cos(180^(@)+theta).sin(270^(@)-theta).cot(270^(@)+theta))=?`

Answer

To solve the expression

\[ \frac{\sin(270^\circ + \theta) \cdot \cos(360^\circ + \theta) \cdot \tan(180^\circ + \theta)}{\cos(180^\circ + \theta) \cdot \sin(270^\circ - \theta) \cdot \cot(270^\circ + \theta)} \]

we will simplify each trigonometric function step by step.

### Step 1: Simplify \(\sin(270^\circ + \theta)\)

Using the sine addition formula, we have:

\[ \sin(270^\circ + \theta) = \sin 270^\circ \cos \theta + \cos 270^\circ \sin \theta \]

Since \(\sin 270^\circ = -1\) and \(\cos 270^\circ = 0\), we get:

\[ \sin(270^\circ + \theta) = -1 \cdot \cos \theta + 0 \cdot \sin \theta = -\cos \theta \]

### Step 2: Simplify \(\cos(360^\circ + \theta)\)

Using the cosine periodicity, we have:

\[ \cos(360^\circ + \theta) = \cos \theta \]

### Step 3: Simplify \(\tan(180^\circ + \theta)\)

Using the tangent addition formula, we have:

\[ \tan(180^\circ + \theta) = \tan \theta \]

### Step 4: Substitute into the numerator

Now substituting these values into the numerator:

\[ \sin(270^\circ + \theta) \cdot \cos(360^\circ + \theta) \cdot \tan(180^\circ + \theta) = (-\cos \theta) \cdot (\cos \theta) \cdot (\tan \theta) = -\cos^2 \theta \tan \theta \]

### Step 5: Simplify \(\cos(180^\circ + \theta)\)

Using the cosine addition formula, we have:

\[ \cos(180^\circ + \theta) = -\cos \theta \]

### Step 6: Simplify \(\sin(270^\circ - \theta)\)

Using the sine subtraction formula, we have:

\[ \sin(270^\circ - \theta) = \sin 270^\circ \cos \theta - \cos 270^\circ \sin \theta = -\cos \theta \]

### Step 7: Simplify \(\cot(270^\circ + \theta)\)

Using the cotangent addition formula, we have:

\[ \cot(270^\circ + \theta) = \frac{1}{\tan(270^\circ + \theta)} = \frac{1}{-\tan \theta} = -\cot \theta \]

### Step 8: Substitute into the denominator

Now substituting these values into the denominator:

\[ \cos(180^\circ + \theta) \cdot \sin(270^\circ - \theta) \cdot \cot(270^\circ + \theta) = (-\cos \theta) \cdot (-\cos \theta) \cdot (-\cot \theta) = -\cos^2 \theta \cot \theta \]

### Step 9: Combine the numerator and denominator

Now we can combine the results:

\[ \frac{-\cos^2 \theta \tan \theta}{-\cos^2 \theta \cot \theta} \]

### Step 10: Simplify the expression

This simplifies to:

\[ \frac{\tan \theta}{\cot \theta} = \tan^2 \theta \]

### Final Answer

Thus, the final answer is:

\[ \tan^2 \theta \]

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