We say an equation `f(x)=g(x)` is consistent, if the curves `y=f(x) and y=g(x)` touch or intersect at atleast one point. If the curves `y=f(x) and y=g(x)` do not intersect or touch, then the equation `f(x)=g(x)` is said to be inconsistent i.e. has no solution.
Among the following equations, which is consistent in `(0, pi//2)`?

Options

A. `sinx+x^(2)=0`
B. `cosx=x`
C. `tanx=x`
D. All of these

Answer

To determine which of the given equations is consistent in the interval \( (0, \frac{\pi}{2}) \), we need to analyze the intersections of the curves represented by the equations. A consistent equation means that the curves intersect or touch at least once within the given interval.

### Step-by-Step Solution:

1. **Identify the equations**:
- The equations provided are:
1. \( f(x) = \sin x \) and \( g(x) = -x^2 \)
2. \( f(x) = \cos x \) and \( g(x) = x \)
3. \( f(x) = \tan x \) and \( g(x) = x \)

2. **Analyze the first equation: \( \sin x = -x^2 \)**:
- The graph of \( y = \sin x \) is positive in the interval \( (0, \frac{\pi}{2}) \).
- The graph of \( y = -x^2 \) is negative in this interval.
- Since one function is always positive and the other is always negative, they do not intersect.
- **Conclusion**: This equation is inconsistent.

3. **Analyze the second equation: \( \cos x = x \)**:
- The graph of \( y = \cos x \) starts at \( 1 \) when \( x = 0 \) and decreases to \( 0 \) as \( x \) approaches \( \frac{\pi}{2} \).
- The graph of \( y = x \) is a straight line starting from \( (0, 0) \) and increasing.
- Since \( \cos x \) decreases from \( 1 \) to \( 0 \) and \( y = x \) increases from \( 0 \) to \( \frac{\pi}{2} \), they must intersect at least once in the interval \( (0, \frac{\pi}{2}) \).
- **Conclusion**: This equation is consistent.

4. **Analyze the third equation: \( \tan x = x \)**:
- The graph of \( y = \tan x \) increases rapidly and approaches infinity as \( x \) approaches \( \frac{\pi}{2} \).
- The graph of \( y = x \) is a straight line.
- They intersect at \( (0, 0) \), but since this point is excluded from the open interval \( (0, \frac{\pi}{2}) \), they do not intersect within the interval.
- **Conclusion**: This equation is inconsistent.

### Final Conclusion:
Among the given equations, the only consistent equation in the interval \( (0, \frac{\pi}{2}) \) is:
- **Option 2: \( \cos x = x \)**