Step-by-step Solution:

1. Simplify the inverse trigonometric terms. Let $\theta = {\cot }^{ - 1}x$. This implies that $\cot \theta = x$. Since the problem states that $0 < x < 1$, the angle $\theta$ must be in the first quadrant, i.e., $0 < \theta < \pi/2$.

2. Construct a right-angled triangle. We can represent $\cot \theta = x = x/1$ using a right-angled triangle where:

   • Adjacent side = $x$
   • Opposite side = $1$
   • The hypotenuse can be calculated using the Pythagorean theorem: \sqrt{{(\text{adjacent})}^2 + {(\text{opposite})}^2}} = \sqrt{x^2 + 1^2} = \sqrt{1 + x^2}.

3. Find the values of $\sin(\cot^{-1}x)$ and $\cos(\cot^{-1}x)$. From the triangle, we can determine the values of $\sin\theta$ and $\cos\theta$:

   • $\sin(\cot^{-1}x) = \sin\theta = {\text{opposite} \over \text{hypotenuse}} = {1 \over \sqrt{1 + x^2}}$
   • $\cos(\cot^{-1}x) = \cos\theta = {\text{adjacent} \over \text{hypotenuse}} = {x \over \sqrt{1 + x^2}}$

4. Substitute these values into the expression inside the curly braces. Let the expression inside the curly braces be $E$. $E = x\cos ( {{{\cot }^{ - 1}}x} ) + \sin ( {{{\cot }^{ - 1}}x} )$ Substituting the values from Step 3: $E = x \cdot \left( {x \over \sqrt{1 + x^2}} \right) + {1 \over \sqrt{1 + x^2}}$ $E = {x^2 \over \sqrt{1 + x^2}} + {1 \over \sqrt{1 + x^2}} = {{x^2 + 1} \over \sqrt{1 + x^2}}$ $E = \sqrt{1 + x^2}$

5. Substitute the simplified expression back into the main expression. The original expression is $\sqrt {1 + {x^2}} {\left[ {{E^2} - 1} \right]^{1/2}}$. Substituting $E = \sqrt{1 + x^2}$: $\sqrt {1 + {x^2}} {\left[ {{{\left( {\sqrt{1 + x^2}} \right)}^2} - 1} \right]^{1/2}}$

6. Simplify the final expression. $= \sqrt {1 + {x^2}} {\left[ {(1 + x^2) - 1} \right]^{1/2}}$ $= \sqrt {1 + {x^2}} {\left[ {x^2} \right]^{1/2}}$ $= \sqrt {1 + {x^2}} \cdot \sqrt{x^2}$ Since it is given that $0 < x < 1$, $x$ is positive. Therefore, $\sqrt{x^2} = |x| = x$. $= \sqrt {1 + {x^2}} \cdot x = x\sqrt {1 + {x^2}}$

7. Compare with the options. The simplified expression is $x\sqrt{1 + x^2}$, which corresponds to option C.

Final Answer:

The correct option is C.