In cyclic quadrilateral ABCD (none of these being `90^(@)`), which of the following is not true ?

Options

A. tan A cot C = -1
B. sec B cos D = -1
C. cosec B sin D = 1
D. none of these

Explanation

To solve the problem regarding the cyclic quadrilateral ABCD and determine which of the statements is not true, we will analyze the properties of cyclic quadrilaterals and the relationships between their angles.

### Step-by-Step Solution:

1. **Understanding Cyclic Quadrilaterals**:
A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. One important property of cyclic quadrilaterals is that the sum of the opposite angles is equal to \(180^\circ\).

\[ \text{If } A, B, C, D \text{ are the angles of the quadrilateral, then:} \]
\[ A + C = 180^\circ \quad \text{and} \quad B + D = 180^\circ \]

**Hint**: Recall the property of cyclic quadrilaterals regarding the sum of opposite angles.

2. **Analyzing the Options**:
We need to check the validity of the statements given in the options. Let's denote the options as follows:
- Option 1: \( \tan A = -\tan C \)
- Option 2: \( \sec B = -\sec D \)
- Option 3: \( \sin B = \sin D \)
- Option 4: (Assumed to be incorrect)

3. **Checking Option 1**:
From the property \( A + C = 180^\circ \):
\[ \tan A = \tan(180^\circ - C) = -\tan C \]
Thus, Option 1 is true.

**Hint**: Use the tangent subtraction identity for angles summing to \(180^\circ\).

4. **Checking Option 2**:
Again, using \( B + D = 180^\circ \):
\[ \sec B = \sec(180^\circ - D) = -\sec D \]
Thus, Option 2 is true.

**Hint**: Recall the secant function's behavior under angle transformations.

5. **Checking Option 3**:
Using \( B + D = 180^\circ \):
\[ \sin B = \sin(180^\circ - D) = \sin D \]
Thus, Option 3 is true.

**Hint**: Use the sine function's property for angles summing to \(180^\circ\).

6. **Conclusion**:
Since Options 1, 2, and 3 are all true, the only option left that is not true must be Option 4.

### Final Answer:
The statement that is not true regarding the cyclic quadrilateral ABCD is **Option 4**.