For `x in(0,1)`, let `alpha=sin^(-1)x,beta=x,gamma=tan^(-1)x, delta=cot^(-1)x-(pi)/(2)`. Which of the following is true ?

Options

A. `alpha gt beta gt gamma`
B. `beta gt alpha gt gamma gt delta`
C. `alpha gt beta gt gamma gt delta`
D. `beta gt alpha gt delta gt gamma`

Explanation

To solve the problem, we need to analyze the relationships between the given inverse trigonometric functions for \( x \in (0, 1) \). Let's define the variables clearly:

- \( \alpha = \sin^{-1}(x) \)
- \( \beta = x \)
- \( \gamma = \tan^{-1}(x) \)
- \( \delta = \cot^{-1}(x) - \frac{\pi}{2} \)

Now, we will evaluate the inequalities step by step.

### Step 1: Analyze \( \alpha \) and \( \beta \)

We know that for \( x \in (0, 1) \):
\[ \sin^{-1}(x) > x \]
This is because the sine function is increasing and \( \sin^{-1}(x) \) gives the angle whose sine is \( x \), which is always greater than \( x \) itself in this interval.

### Step 2: Analyze \( \beta \) and \( \gamma \)

Next, we analyze the relationship between \( \beta \) and \( \gamma \):
\[ x > \tan^{-1}(x) \]
For \( x \in (0, 1) \), \( \tan^{-1}(x) \) is less than \( x \) because the tangent function grows slower than the linear function \( y = x \) in this interval.

### Step 3: Analyze \( \gamma \) and \( \delta \)

Now we need to analyze \( \gamma \) and \( \delta \):
\[ \tan^{-1}(x) > \cot^{-1}(x) - \frac{\pi}{2} \]
Using the identity \( \cot^{-1}(x) - \frac{\pi}{2} = -\tan^{-1}(x) \), we can rewrite this as:
\[ \tan^{-1}(x) > -\tan^{-1}(x) \]
This simplifies to:
\[ 2\tan^{-1}(x) > 0 \]
which is true for \( x > 0 \).

### Step 4: Combine the inequalities

From the steps above, we have established the following inequalities:
1. \( \alpha > \beta \)
2. \( \beta > \gamma \)
3. \( \gamma > \delta \)

Thus, we can conclude:
\[ \alpha > \beta > \gamma > \delta \]

### Conclusion

The correct order of the functions from greatest to least is:
\[ \alpha > \beta > \gamma > \delta \]

### Final Answer

The correct option is \( C \).