If `theta` lies in the first quadrant which of the following in not true

Options

A. `(theta)/(2)lttan ((theta)/(2))`
B. `(theta)/(2)ltsin ""(theta)/(2)`
C. `thetacos^(2)((theta)/(2))ltsintheta`
D. `theta sin"(theta)/(2)lt 2 sin""(theta)/(2)`

Answer

To solve the problem, we need to determine which of the statements regarding the angle `theta` in the first quadrant is not true.

### Step-by-Step Solution:

1. **Understanding the Quadrant**:
Since `theta` lies in the first quadrant, we know that:
\[ 0 \leq \theta \leq \frac{\pi}{2} \]

2. **Define the Function**:
Let's define a function based on the problem:
\[ f(\theta) = \frac{\theta}{2} - \sin\left(\frac{\theta}{2}\right) \]

3. **Differentiate the Function**:
We differentiate `f(θ)` with respect to `θ`:
\[ f'(\theta) = \frac{1}{2} - \cos\left(\frac{\theta}{2}\right) \cdot \frac{1}{2} \]
Simplifying this gives:
\[ f'(\theta) = \frac{1}{2} - \frac{1}{2} \cos\left(\frac{\theta}{2}\right) \]

4. **Analyze the Derivative**:
Since `cos(θ/2)` can take values from 1 to 0 in the first quadrant (as `θ` goes from 0 to π/2), we can see that:
\[ f'(\theta) = \frac{1}{2} - \frac{1}{2} \cos\left(\frac{\theta}{2}\right) > 0 \]
This means that `f(θ)` is an increasing function in the interval \(0 \leq \theta \leq \frac{\pi}{2}\).

5. **Evaluate at the Endpoints**:
Now, we evaluate `f(0)`:
\[ f(0) = \frac{0}{2} - \sin\left(\frac{0}{2}\right) = 0 - 0 = 0 \]
Since `f(θ)` is increasing, for any `θ` in the interval \(0 < θ \leq \frac{\pi}{2}\):
\[ f(\theta) > f(0) = 0 \]

6. **Conclude the Inequality**:
Thus, we conclude that:
\[ \frac{\theta}{2} - \sin\left(\frac{\theta}{2}\right) > 0 \implies \frac{\theta}{2} > \sin\left(\frac{\theta}{2}\right) \]
This means that:
\[ \frac{\theta}{2} > \sin\left(\frac{\theta}{2}\right) \]

7. **Identify the False Statement**:
If one of the options states that:
\[ \frac{\theta}{2} < \sin\left(\frac{\theta}{2}\right) \]
This is not true based on our analysis. Therefore, this option is the correct answer to the question.

### Conclusion:
The option that states \(\frac{\theta}{2} < \sin\left(\frac{\theta}{2}\right)\) is not true when `theta` lies in the first quadrant.