What is the value of $\{(1-\sin^2 heta)\sec^2 heta + an^2 heta(\cos^2 heta+1)\}$, where $0^\circ

Q: What is the value of $\{(1-\sin^2\theta)\sec^2\theta + \tan^2\theta(\cos^2\theta+1)\}$, where $0^\circ < \theta < 90^\circ$ is :

What is the value of $\{(1-\sin^2\theta)\sec^2\theta + \tan^2\theta(\cos^2\theta+1)\}$, where $0^\circ < \theta < 90^\circ$ is :

Question

What is the value of $\{(1-\sin^2	heta)\sec^2	heta + 	an^2	heta(\cos^2	heta+1)\}$, where $0^\circ < 	heta < 90^\circ$ is :

Accepted Answer

To determine the value of the expression $\{(1-\sin^2\theta)\sec^2\theta + \tan^2\theta(\cos^2\theta+1)\}$, where $0^\circ < \theta < 90^\circ$, let's analyze it step by step.

### Step-by-Step Explanation:

1. **Simplify $1 - \sin^2\theta$:**
   $$
   1 - \sin^2\theta = \cos^2\theta
   $$
   This is a fundamental trigonometric identity.

2. **Substitute into the expression:**
   $$
   \cos^2\theta \cdot \sec^2\theta + \tan^2\theta(\cos^2\theta + 1)
   $$

3. **Simplify $\cos^2\theta \cdot \sec^2\theta$:**
   $$
   \cos^2\theta \cdot \sec^2\theta = \cos^2\theta \cdot \frac{1}{\cos^2\theta} = 1
   $$

4. **Simplify $\tan^2\theta(\cos^2\theta + 1)$:**
   $$
   \tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}
   $$
   Therefore,
   $$
   \tan^2\theta(\cos^2\theta + 1) = \frac{\sin^2\theta}{\cos^2\theta}(\cos^2\theta + 1) = \sin^2\theta + \frac{\sin^2\theta}{\cos^2\theta}
   $$

5. **Combine the terms:**
   $$
   1 + \sin^2\theta + \frac{\sin^2\theta}{\cos^2\theta}
   $$

6. **Analyze the expression:**
   - The term $\sin^2\theta$ is always positive and less than 1 for $0^\circ < \theta < 90^\circ$.
   - The term $\frac{\sin^2\theta}{\cos^2\theta} = \tan^2\theta$ is also positive and increases as $\theta$ approaches $90^\circ$.

7. **Conclusion:**
   - The expression $1 + \sin^2\theta + \tan^2\theta$ is greater than 2 because:
     - $1 + \sin^2\theta$ is greater than 1.
     - $\tan^2\theta$ is positive and adds to the total, making the entire expression greater than 2.

### Justification for Option 2:

- **Option 2 (> 2):** The expression is indeed greater than 2 due to the positive contributions from both $\sin^2\theta$ and $\tan^2\theta$.

### Explanation for Incorrect Options:

- **Option 1 (2):** The expression cannot be exactly 2 because both $\sin^2\theta$ and $\tan^2\theta$ contribute positive values that exceed 1 when added to the initial 1.

- **Option 3 (< 2):** The expression cannot be less than 2 because the sum of positive terms $\sin^2\theta$ and $\tan^2\theta$ ensures the total is greater than 2.

- **Option 4 (None of the above):** Since Option 2 is correct, this option is not applicable.

Thus, the correct answer is indeed Option 2, as the expression is greater than 2 for $0^\circ < \theta < 90^\circ$.