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

Options

A. `sin theta`
B. `cos theta`
C. 1
D. `(1)/(sqrt2)`

Answer

To solve the expression

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

we will evaluate each trigonometric function step by step.

### Step 1: Simplify \(\sin(90^\circ - \theta)\)

Using the co-function identity, we know that:

\[ \sin(90^\circ - \theta) = \cos(\theta). \]

### Step 2: Simplify \(\sec(180^\circ - \theta)\)

The secant function is the reciprocal of cosine. Thus:

\[ \sec(180^\circ - \theta) = \frac{1}{\cos(180^\circ - \theta)}. \]

Since \(\cos(180^\circ - \theta) = -\cos(\theta)\), we have:

\[ \sec(180^\circ - \theta) = -\frac{1}{\cos(\theta)}. \]

### Step 3: Simplify \(\sin(-\theta)\)

Using the odd function property of sine, we find:

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

### Step 4: Substitute into the numerator

Now substituting these results into the numerator:

\[ \sin(90^\circ - \theta) \sec(180^\circ - \theta) \sin(-\theta) = \cos(\theta) \left(-\frac{1}{\cos(\theta)}\right)(-\sin(\theta)). \]

This simplifies to:

\[ \cos(\theta) \left(-\frac{1}{\cos(\theta)}\right)(-\sin(\theta)) = \sin(\theta). \]

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

Using the sine addition formula, we have:

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

### Step 6: Simplify \(\cot(360^\circ - \theta)\)

Using the cotangent identity, we find:

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

### Step 7: Simplify \(\csc(90^\circ + \theta)\)

Using the cosecant identity, we have:

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

### Step 8: Substitute into the denominator

Now substituting these results into the denominator:

\[ \sin(180^\circ + \theta) \cot(360^\circ - \theta) \csc(90^\circ + \theta) = (-\sin(\theta))(-\cot(\theta))(\sec(\theta)). \]

This simplifies to:

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

### Step 9: Final expression

Now the entire expression becomes:

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

### Step 10: Simplifying the expression

Since \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we can rewrite the denominator:

\[ \sin(\theta) \cdot \frac{\cos(\theta)}{\sin(\theta)} \cdot \frac{1}{\cos(\theta)} = 1. \]

Thus, the expression simplifies to:

\[ \frac{\sin(\theta)}{1} = \sin(\theta). \]

### Conclusion

The value of the expression is:

\[ \sin(\theta). \]

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