Write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of $\theta$ only. \[ \frac{1-\csc (-\theta)}{\csc (-\theta)} \] \[ \frac{1-\csc (-\theta)}{\csc (-\theta)}= \]

Step 1 ：Write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. The given expression is \(\frac{1-\csc (-\theta)}{\csc (-\theta)}\)

Step 2 ：The first step is to convert the cosecant function to its equivalent in terms of sine. The cosecant of an angle is the reciprocal of the sine of the angle. Therefore, we can rewrite the expression as follows: \(\frac{1-\frac{1}{\sin(-\theta)}}{\frac{1}{\sin(-\theta)}}\)

Step 3 ：Next, we can simplify the expression by multiplying the numerator and the denominator by \(\sin(-\theta)\) to eliminate the fractions. This will give us: \(\frac{\sin(-\theta)-1}{1}\)

Step 4 ：Finally, we can simplify the expression further by noting that \(\sin(-\theta) = -\sin(\theta)\). This gives us the final expression: \(-\sin(\theta)-1\)

Step 5 ：The simplified expression in terms of sine and cosine, with no quotients and all functions of \(\theta\) only, is \(\boxed{1 + \frac{1}{\sin(\theta)}}\)