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

Solution :
$$\eqalign{ & {\text{Let, }}\,{\sin ^{ - 1}}a = x \cr & \therefore a = \sin x \cr & {\sin ^{ - 1}}b = y \cr & \therefore b = \sin y;{\sin ^{ - 1}}c = z \cr & \therefore c = \sin z \cr & \therefore a\sqrt {1 - {a^2}} + b\sqrt {1 - {b^2}} + c\sqrt {1 - {c^2}} \cr & = \sin x\cos x + \sin y\cos y + \sin z\cos z \cr & = \left( {\frac{1}{2}} \right)\left( {\sin 2x + \sin 2y + \sin 2z} \right) = \left( {\frac{1}{2}} \right)\left( {4\sin x\sin y\sin z} \right) \cr & = 2\sin x\sin y\sin z = 2abc \cr} $$