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

Options

A. `sintheta`
B. `costheta`
C. 1
D. `1/sqrt2`

Answer

To solve the expression

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

we will simplify each term step by step.

### Step 1: Simplify \(\sin(90^\circ - \theta)\)
Using the co-function identity, we have:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]

### Step 2: Simplify \(\sec(180^\circ - \theta)\)
In the second quadrant, the secant function is negative:
\[ \sec(180^\circ - \theta) = -\sec(\theta) \]

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

### Step 4: Substitute into the numerator
Now, substituting back into the numerator:
\[ \text{Numerator} = \cos(\theta) \cdot (-\sec(\theta)) \cdot (-\sin(\theta)) = \cos(\theta) \cdot \sec(\theta) \cdot \sin(\theta) \]
Since \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we have:
\[ \text{Numerator} = \sin(\theta) \]

### Step 5: Simplify \(\sin(180^\circ + \theta)\)
In the third quadrant, sine is negative:
\[ \sin(180^\circ + \theta) = -\sin(\theta) \]

### Step 6: Simplify \(\cot(360^\circ - \theta)\)
In the fourth quadrant, cotangent is positive:
\[ \cot(360^\circ - \theta) = \cot(\theta) \]

### Step 7: Simplify \(\csc(90^\circ + \theta)\)
In the first quadrant, cosecant is positive:
\[ \csc(90^\circ + \theta) = \sec(\theta) \]

### Step 8: Substitute into the denominator
Now, substituting back into the denominator:
\[ \text{Denominator} = (-\sin(\theta)) \cdot \cot(\theta) \cdot \sec(\theta) \]
Using \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we have:
\[ \text{Denominator} = -\sin(\theta) \cdot \frac{\cos(\theta)}{\sin(\theta)} \cdot \frac{1}{\cos(\theta)} = -1 \]

### Step 9: Combine numerator and denominator
Now we can combine the numerator and denominator:
\[ \frac{\sin(\theta)}{-1} = -\sin(\theta) \]

### Final Result
Thus, the value of the original expression is:
\[ -\sin(\theta) \]