`If 0 lt x lt pi /2 ` then

Options

A. `tan x lt x lt sin x`
B. `x lt sin x lt tan x `
C. `sin x lt tan x lt x 1`
D. none of these

Explanation

To solve the problem, we need to analyze the functions \( \tan x \), \( x \), and \( \sin x \) in the interval \( 0 < x < \frac{\pi}{2} \). We will determine their relationships by comparing their values and analyzing their graphs.

### Step 1: Understand the Functions
We are looking at three functions:
1. \( \tan x \)
2. \( x \)
3. \( \sin x \)

### Step 2: Analyze the Interval
The interval given is \( 0 < x < \frac{\pi}{2} \). In this interval:
- \( \tan x \) is increasing and approaches infinity as \( x \) approaches \( \frac{\pi}{2} \).
- \( \sin x \) is also increasing and approaches 1 as \( x \) approaches \( \frac{\pi}{2} \).
- The line \( y = x \) is a straight line that increases linearly.

### Step 3: Graph the Functions
To visualize the relationships:
- Draw the graph of \( \sin x \) which starts at (0,0) and increases to (π/2, 1).
- Draw the graph of \( \tan x \) which starts at (0,0) and increases rapidly, approaching infinity as \( x \) approaches \( \frac{\pi}{2} \).
- Draw the line \( y = x \) which is a diagonal line from (0,0) to (π/2, π/2).

### Step 4: Compare the Functions
In the interval \( 0 < x < \frac{\pi}{2} \):
- At \( x = 0 \), \( \tan(0) = 0 \), \( \sin(0) = 0 \), and \( x = 0 \).
- As \( x \) increases, \( \tan x \) increases faster than \( \sin x \) and the line \( y = x \).
- We can use derivatives to confirm this:
- The derivative of \( \tan x \) is \( \sec^2 x \), which is always positive in the interval.
- The derivative of \( \sin x \) is \( \cos x \), which is also positive.
- The derivative of \( x \) is 1, which is constant.

### Step 5: Establish the Relationships
From the graphs and the analysis:
- \( \tan x > x \) for \( 0 < x < \frac{\pi}{2} \)
- \( x > \sin x \) for \( 0 < x < \frac{\pi}{2} \)

Thus, we conclude:
\[ \tan x > x > \sin x \quad \text{for} \quad 0 < x < \frac{\pi}{2} \]

### Final Answer
The correct relationship is:
\[ \tan x > x > \sin x \]