What is the value of `(tan (90^@-theta) sec (180^@-theta)sin(-theta))/(sin (180^@+theta) cot (360^@-theta) "cosec"(90^@-theta))` ?

Answer

To solve the expression

\[ \frac{\tan(90^\circ - \theta) \cdot \sec(180^\circ - \theta) \cdot \sin(-\theta)}{\sin(180^\circ + \theta) \cdot \cot(360^\circ - \theta) \cdot \csc(90^\circ - \theta)} \]

we will simplify each trigonometric function step by step.

### Step 1: Simplify \(\tan(90^\circ - \theta)\)
Using the co-function identity, we know that:

\[ \tan(90^\circ - \theta) = \cot(\theta) \]

### Step 2: Simplify \(\sec(180^\circ - \theta)\)
Using the identity for secant in the second quadrant:

\[ \sec(180^\circ - \theta) = -\sec(\theta) \]

### Step 3: Simplify \(\sin(-\theta)\)
Using the odd function property of sine:

\[ \sin(-\theta) = -\sin(\theta) \]

### Step 4: Simplify \(\sin(180^\circ + \theta)\)
Using the sine function property in the third quadrant:

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

### Step 5: Simplify \(\cot(360^\circ - \theta)\)
Using the cotangent property in the fourth quadrant:

\[ \cot(360^\circ - \theta) = \cot(\theta) \]

### Step 6: Simplify \(\csc(90^\circ - \theta)\)
Using the co-function identity for cosecant:

\[ \csc(90^\circ - \theta) = \sec(\theta) \]

### Step 7: Substitute the simplified values into the expression
Now substituting all simplified values back into the original expression:

\[ \frac{\cot(\theta) \cdot (-\sec(\theta)) \cdot (-\sin(\theta))}{(-\sin(\theta)) \cdot \cot(\theta) \cdot \sec(\theta)} \]

### Step 8: Simplify the expression
The numerator becomes:

\[ \cot(\theta) \cdot \sec(\theta) \cdot \sin(\theta) \]

The denominator becomes:

\[ \sin(\theta) \cdot \cot(\theta) \cdot \sec(\theta) \]

Thus, the expression simplifies to:

\[ \frac{\cot(\theta) \cdot \sec(\theta) \cdot \sin(\theta)}{\sin(\theta) \cdot \cot(\theta) \cdot \sec(\theta)} = 1 \]

### Final Answer
The value of the expression is:

\[ \boxed{1} \]