Simplify the following: (i)`"sin"(90^(@)+theta)"tan"(270^(@)+theta)cot(90^(@)+theta) cosec(270^(@)+theta)`
(ii) `("sin"(-theta)"tan"(180^(@)+theta)"tan"(90^(@)+theta))/(cot(90^(@)-theta)cos(360^(@)-theta)"sin"(180^(@)-theta))`

Answer

Let's simplify the given expressions step by step.

### (i) Simplifying `sin(90° + θ) * tan(270° + θ) * cot(90° + θ) * cosec(270° + θ)`

1. **Evaluate `sin(90° + θ)`**:
- Using the identity: `sin(90° + θ) = cos(θ)`.
- So, `sin(90° + θ) = cos(θ)`.

2. **Evaluate `tan(270° + θ)`**:
- Using the identity: `tan(270° + θ) = cot(θ)`.
- So, `tan(270° + θ) = cot(θ)`.

3. **Evaluate `cot(90° + θ)`**:
- Using the identity: `cot(90° + θ) = -tan(θ)`.
- So, `cot(90° + θ) = -tan(θ)`.

4. **Evaluate `cosec(270° + θ)`**:
- Using the identity: `cosec(270° + θ) = -sec(θ)`.
- So, `cosec(270° + θ) = -sec(θ)`.

5. **Combine all the results**:
- Now substituting back into the expression:
\[ cos(θ) * cot(θ) * (-tan(θ)) * (-sec(θ)) \]
- This simplifies to:
\[ cos(θ) * cot(θ) * tan(θ) * sec(θ) \]

6. **Simplify further**:
- Since `cot(θ) * tan(θ) = 1` and `cos(θ) * sec(θ) = 1`, we have:
\[ 1 * 1 = 1 \]

7. **Final result**:
- The simplified expression is:
\[ -1 \]

### (ii) Simplifying `sin(-θ) * tan(180° + θ) * tan(90° + θ) / (cot(90° - θ) * cos(360° - θ) * sin(180° - θ))`

1. **Evaluate `sin(-θ)`**:
- Using the identity: `sin(-θ) = -sin(θ)`.
- So, `sin(-θ) = -sin(θ)`.

2. **Evaluate `tan(180° + θ)`**:
- Using the identity: `tan(180° + θ) = tan(θ)`.
- So, `tan(180° + θ) = tan(θ)`.

3. **Evaluate `tan(90° + θ)`**:
- Using the identity: `tan(90° + θ) = -cot(θ)`.
- So, `tan(90° + θ) = -cot(θ)`.

4. **Evaluate `cot(90° - θ)`**:
- Using the identity: `cot(90° - θ) = tan(θ)`.
- So, `cot(90° - θ) = tan(θ)`.

5. **Evaluate `cos(360° - θ)`**:
- Using the identity: `cos(360° - θ) = cos(θ)`.
- So, `cos(360° - θ) = cos(θ)`.

6. **Evaluate `sin(180° - θ)`**:
- Using the identity: `sin(180° - θ) = sin(θ)`.
- So, `sin(180° - θ) = sin(θ)`.

7. **Combine all the results**:
- Now substituting back into the expression:
\[ (-sin(θ)) * (tan(θ)) * (-cot(θ)) / (tan(θ) * cos(θ) * sin(θ)) \]
- This simplifies to:
\[ (sin(θ) * tan(θ) * cot(θ)) / (tan(θ) * cos(θ) * sin(θ)) \]

8. **Simplify further**:
- Since `tan(θ) * cot(θ) = 1` and `sin(θ)` cancels out:
\[ 1 / cos(θ) \]

9. **Final result**:
- The simplified expression is:
\[ sec(θ) \]

### Summary of Results:
- (i) The simplified expression is `-1`.
- (ii) The simplified expression is `sec(θ)`.