Which one of the following is true for `0^(@)ltthetalt90^(@)`?

Options

A. `costhetalecos^(2)theta`
B. `costhetagtcos^(2)theta`
C. `costhetaltcos^(2)theta`
D. `costhetagecos^(2)theta`

Answer

To solve the problem of determining the relationship between \(\cos \theta\) and \(\cos^2 \theta\) for \(0^\circ < \theta < 90^\circ\), we can follow these steps:

### Step 1: Understand the Functions
We need to analyze the functions \(\cos \theta\) and \(\cos^2 \theta\). The cosine function decreases from 1 to 0 as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).

### Step 2: Evaluate at Specific Angles
Let's evaluate both \(\cos \theta\) and \(\cos^2 \theta\) at some key angles within the interval \(0^\circ < \theta < 90^\circ\):
- For \(\theta = 0^\circ\):
- \(\cos 0^\circ = 1\)
- \(\cos^2 0^\circ = 1^2 = 1\)

- For \(\theta = 30^\circ\):
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)

- For \(\theta = 45^\circ\):
- \(\cos 45^\circ = \frac{1}{\sqrt{2}} \approx 0.707\)
- \(\cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)

- For \(\theta = 60^\circ\):
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)

### Step 3: Compare Values
Now, we compare the values of \(\cos \theta\) and \(\cos^2 \theta\) for each angle:
- At \(0^\circ\): \(1 = 1\)
- At \(30^\circ\): \(\frac{\sqrt{3}}{2} \approx 0.866 > \frac{3}{4} = 0.75\)
- At \(45^\circ\): \(\frac{1}{\sqrt{2}} \approx 0.707 > \frac{1}{2} = 0.5\)
- At \(60^\circ\): \(\frac{1}{2} = 0.5 > \frac{1}{4} = 0.25\)

### Step 4: General Conclusion
From the evaluations, we can see that for \(0^\circ < \theta < 60^\circ\), \(\cos \theta > \cos^2 \theta\). At \(60^\circ\), they are equal, and for \(60^\circ < \theta < 90^\circ\), \(\cos \theta < \cos^2 \theta\).

Thus, the relationship we are looking for is:
\[ \cos \theta > \cos^2 \theta \text{ for } 0^\circ < \theta < 60^\circ \]

### Final Answer
The true statement for \(0^\circ < \theta < 90^\circ\) is:
\[ \cos \theta > \cos^2 \theta \text{ for } 0^\circ < \theta < 60^\circ \]