Select the INCORRECT match.

Options

A. `cos^(2)theta+(1)/(1+cot^(2)theta)=1`
B. `(1+tan^(2)theta)(1+sintheta)(1-sintheta)=1`
C. `(tantheta+sintheta)/(tantheta-sintheta)=(sectheta-1)/(sectheta+1)`
D. `(sin^(3)theta+cos^(3)theta)/(sintheta+costheta)+sinthetacostheta=1`

Answer

To solve the question of selecting the INCORRECT match from the given options, we will analyze each option step by step.

### Step-by-Step Solution:

**Step 1: Analyze Option 1**
- The expression is \( \cos^2 \theta + \frac{1}{1 + \cot^2 \theta} \).
- We know that \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \).
- Therefore, \( 1 + \cot^2 \theta = 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} \) (using \( \sin^2 \theta + \cos^2 \theta = 1 \)).
- Thus, \( \frac{1}{1 + \cot^2 \theta} = \sin^2 \theta \).
- Now, substituting back, we have \( \cos^2 \theta + \sin^2 \theta = 1 \).
- **Conclusion**: Option 1 is correct.

**Step 2: Analyze Option 2**
- The expression is \( 1 + \tan^2 \theta \).
- We know that \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \).
- Therefore, \( 1 + \tan^2 \theta = 1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \).
- **Conclusion**: Option 2 is correct.

**Step 3: Analyze Option 3**
- The expression is \( \tan \theta + \sin \theta \).
- We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
- Therefore, \( \tan \theta + \sin \theta = \frac{\sin \theta}{\cos \theta} + \sin \theta = \frac{\sin \theta + \sin \theta \cos \theta}{\cos \theta} = \frac{\sin \theta (1 + \cos \theta)}{\cos \theta} \).
- This does not simplify to a standard identity.
- **Conclusion**: Option 3 is incorrect.

**Step 4: Analyze Option 4**
- The expression is \( \frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} \).
- We can apply the identity for the sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).
- Here, \( a = \sin \theta \) and \( b = \cos \theta \).
- Thus, \( \sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta) \).
- The denominator \( \sin \theta + \cos \theta \) cancels out, leaving \( \sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta = 1 - \sin \theta \cos \theta \).
- **Conclusion**: Option 4 is correct.

### Final Conclusion:
- The incorrect match is **Option 3**.