If ${\sin }^{-1}a+{\sin }^{-1}b+{\sin }^{-1}c=\pi$, then the value of $a\sqrt{1-a^2}+b\sqrt{1-b^2}+c\sqrt{1-c^2}$ will be:

Let $A={\sin }^{-1}a,B={\sin }^{-1}b,C={\sin }^{-1}c$.Then $a=\sin A,b=\sin B,c=\sin C$ and $A+B+C=\pi$.The given expression is:
$$\sin A\cos A+\sin B\cos B+\sin C\cos C$$
Multiply and divide by 2:
$$\frac{1}{2}(\sin 2A+\sin 2B+\sin 2C)$$
Using the trigonometric identity for $A+B+C=\pi$:
$$\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$$
Substituting this into our expression:
$$\frac{1}{2}(4\sin A\sin B\sin C)=2\sin A\sin B\sin C=2abc$$