Find the range of each of the following functions
`sec((pi)/(4)cos^(2)x),ooltxltoo`

Answer

To find the range of the function \( f(x) = \sec\left(\frac{\pi}{4} \cos^2 x\right) \), we will follow these steps:

### Step 1: Determine the range of \( \cos^2 x \)

The cosine function, \( \cos x \), varies between -1 and 1 for all \( x \). Therefore, when we square \( \cos x \), the range of \( \cos^2 x \) will be:

\[ 0 \leq \cos^2 x \leq 1 \]

### Step 2: Multiply the range by \( \frac{\pi}{4} \)

Next, we will multiply the entire range of \( \cos^2 x \) by \( \frac{\pi}{4} \):

\[ 0 \leq \frac{\pi}{4} \cos^2 x \leq \frac{\pi}{4} \]

### Step 3: Define \( \theta \)

Let \( \theta = \frac{\pi}{4} \cos^2 x \). From the previous step, we have:

\[ 0 \leq \theta \leq \frac{\pi}{4} \]

### Step 4: Find the range of \( \sec \theta \)

Now, we need to find the range of \( \sec \theta \) as \( \theta \) varies from \( 0 \) to \( \frac{\pi}{4} \). The secant function is defined as:

\[ \sec \theta = \frac{1}{\cos \theta} \]

The values of \( \cos \theta \) in the interval \( [0, \frac{\pi}{4}] \) will range from \( 1 \) (at \( \theta = 0 \)) to \( \frac{1}{\sqrt{2}} \) (at \( \theta = \frac{\pi}{4} \)). Therefore, the range of \( \sec \theta \) will be:

\[ \sec(0) = 1 \quad \text{and} \quad \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \]

### Step 5: Determine the range of \( f(x) \)

Thus, as \( \theta \) varies from \( 0 \) to \( \frac{\pi}{4} \), \( \sec \theta \) varies from \( 1 \) to \( \sqrt{2} \). Therefore, the range of the function \( f(x) = \sec\left(\frac{\pi}{4} \cos^2 x\right) \) is:

\[ [1, \sqrt{2}] \]

### Final Answer

The range of the function \( f(x) = \sec\left(\frac{\pi}{4} \cos^2 x\right) \) is:

\[ [1, \sqrt{2}] \]