For `0ltxlt(pi)/(2), (1+4cosecx)(1+8secx)`, is

Options

A. `ge81`
B. `gt81`
C. `gt83`
D. `ge83`

Explanation

To solve the inequality \( (1 + 4 \csc x)(1 + 8 \sec x) \) for \( 0 < x < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the expressions
We start by rewriting the cosecant and secant functions in terms of sine and cosine:
\[ \csc x = \frac{1}{\sin x} \quad \text{and} \quad \sec x = \frac{1}{\cos x} \]
Thus, we can rewrite the expression as:
\[ (1 + 4 \csc x)(1 + 8 \sec x) = \left(1 + \frac{4}{\sin x}\right)\left(1 + \frac{8}{\cos x}\right) \]

### Step 2: Simplify the expression
Now, we can expand the expression:
\[ = 1 + \frac{4}{\sin x} + \frac{8}{\cos x} + \frac{32}{\sin x \cos x} \]

### Step 3: Combine terms
We can combine the terms into a single fraction:
\[ = \frac{\sin x \cos x + 4 \cos x + 8 \sin x + 32}{\sin x \cos x} \]

### Step 4: Use the identity for sine and cosine
Recall the identity \( 2 \sin x \cos x = \sin 2x \):
\[ = \frac{2 \sin x \cos x + 8 \sin x + 4 \cos x + 32}{\sin x \cos x} \]

### Step 5: Find the maximum of the denominator
To minimize the entire expression, we need to maximize the denominator \( \sin x \cos x \). The maximum value occurs at \( x = \frac{\pi}{4} \):
\[ \sin x \cos x = \frac{1}{2} \quad \text{when } x = \frac{\pi}{4} \]

### Step 6: Substitute \( x = \frac{\pi}{4} \)
Now, substituting \( x = \frac{\pi}{4} \):
\[ \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
Thus,
\[ = \frac{2 \cdot \frac{1}{2} + 8 \cdot \frac{1}{\sqrt{2}} + 4 \cdot \frac{1}{\sqrt{2}} + 32}{\frac{1}{2}} \]
\[ = \frac{1 + 8\sqrt{2} + 4\sqrt{2} + 32}{\frac{1}{2}} \]
\[ = 2(1 + 12\sqrt{2} + 32) \]
\[ = 2(33 + 12\sqrt{2}) \]

### Step 7: Calculate the value
Now we can calculate the approximate value:
\[ = 66 + 24\sqrt{2} \]
Using \( \sqrt{2} \approx 1.414 \):
\[ = 66 + 24 \times 1.414 \approx 66 + 33.936 \approx 99.936 \]

### Conclusion
Thus, the value of \( (1 + 4 \csc x)(1 + 8 \sec x) \) is approximately \( 99.936 \), which is greater than 81.