Let `0ltalpha,beta,gammalt(pi)/2` are the solutions of the equations `cosx=x,cos(sinx)=x` and `sin(cosx)=x` respectively, then show that `gammaltalphaltbeta`.

Explanation

To show that \( \gamma < \alpha < \beta \), where \( \alpha, \beta, \gamma \) are the solutions of the equations \( \cos x = x \), \( \cos(\sin x) = x \), and \( \sin(\cos x) = x \) respectively, we can analyze the graphs of these functions step by step.

### Step 1: Analyze the first equation \( \cos x = x \)

1. **Graph the functions**: Plot \( y = \cos x \) and \( y = x \).
2. **Intersection**: The solution \( \alpha \) is the x-coordinate of the intersection point of these two graphs.
3. **Behavior of the functions**:
- The function \( y = \cos x \) starts at \( (0, 1) \) and decreases to \( (0, 0) \) as \( x \) approaches \( \frac{\pi}{2} \).
- The line \( y = x \) starts at \( (0, 0) \) and increases linearly.

From the graphs, we can see that \( \alpha \) is the point where \( \cos x \) intersects \( y = x \) in the interval \( (0, \frac{\pi}{2}) \).

### Step 2: Analyze the second equation \( \cos(\sin x) = x \)

1. **Graph the functions**: Plot \( y = \cos(\sin x) \) and \( y = x \).
2. **Intersection**: The solution \( \beta \) is the x-coordinate of the intersection point of these two graphs.
3. **Behavior of the functions**:
- The function \( y = \cos(\sin x) \) oscillates between \( \cos(0) = 1 \) and \( \cos(1) \) as \( x \) increases.
- The line \( y = x \) continues to increase.

Since \( \cos(\sin x) \) starts at 1 and decreases, it will intersect the line \( y = x \) at some point \( \beta \) which is greater than \( \alpha \).

### Step 3: Analyze the third equation \( \sin(\cos x) = x \)

1. **Graph the functions**: Plot \( y = \sin(\cos x) \) and \( y = x \).
2. **Intersection**: The solution \( \gamma \) is the x-coordinate of the intersection point of these two graphs.
3. **Behavior of the functions**:
- The function \( y = \sin(\cos x) \) starts at \( \sin(1) \) when \( x = 0 \) and oscillates as \( x \) increases.
- The line \( y = x \) continues to increase.

From the graph, we can see that \( \gamma \) is less than both \( \alpha \) and \( \beta \).

### Conclusion

From the analysis of the graphs:
- We have established that \( \beta > \alpha \) because the line \( y = x \) intersects \( y = \cos(\sin x) \) at a point greater than where it intersects \( y = \cos x \).
- We have also established that \( \alpha > \gamma \) because \( \gamma \) is the intersection of \( y = \sin(\cos x) \) which is below the intersection of \( y = \cos x \).

Thus, we conclude that:
\[ \gamma < \alpha < \beta \]