Which of the following is not an identity ?

Options

A. `(1)/( sec A - 1) + (1)/( sec A + 1)` = 2 cosec A cot A
B. `cos^(2) A + (1)/( 1 + cot^(2) A ) = 1 `
C. ` tan theta - cot theta = (2 sin^(2) theta - 1)/( sin theta cos theta )`
D. `cot theta + tan theta = cosec theta *cos theta `

Answer

To determine which of the given options is not an identity, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Option 1: \( \frac{1}{\sec A - 1} + \frac{1}{\sec A + 1} \)**

1. Combine the fractions:
\[ \frac{1}{\sec A - 1} + \frac{1}{\sec A + 1} = \frac{(\sec A + 1) + (\sec A - 1)}{(\sec A - 1)(\sec A + 1)} \]
This simplifies to:
\[ \frac{2\sec A}{\sec^2 A - 1} \]

2. Use the identity \( \sec^2 A - 1 = \tan^2 A \):
\[ = \frac{2\sec A}{\tan^2 A} \]

3. Rewrite \( \tan^2 A \) as \( \frac{\sin^2 A}{\cos^2 A} \):
\[ = \frac{2\sec A \cdot \cos^2 A}{\sin^2 A} = \frac{2\cos A}{\sin^2 A} \]

4. This expression is equal to \( 2 \cot A \), which is valid.