`"For x " in (0,(pi)/(2)) , "Prove that sin "x gt x -(x^(2))/(6)`

Explanation

To prove that \( \sin x > x - \frac{x^3}{6} \) for \( x \in (0, \frac{\pi}{2}) \), we can define a function and analyze its behavior. Here’s the step-by-step solution:

### Step 1: Define the function
Let
\[ f(x) = \sin x - x + \frac{x^3}{6} \]
We want to show that \( f(x) > 0 \) for \( x \in (0, \frac{\pi}{2}) \).
...