If \overrightarrow { \mathrm { u } } , \mathrm { v } ^ { \rightarrow } , \mathrm { w } ^ { \rightarrow } are three noncoplanar unit vectors and $\alpha , \beta , \gamma$ are the angles between $\mathrm { u } \rightarrow$ are and $\mathrm { v } ^ { \rightarrow } , \mathrm { v } ^ { \rightarrow } are and $\overrightarrow { \mathrm { w } } , \mathrm { w } ^ { \rightarrow } and $\overrightarrow { \mathrm { u } }$ are respectively and $\overrightarrow { \mathrm { x } } , \mathrm { y } ^ { \rightarrow } , \mathrm { z }$ are unit vectors along the bisectors of the angles $\mathrm { a } , \mathrm { b } , \mathrm { g }$ respectively. Prove that $$\left[ \begin{array} { l l l } \vec { x } \times \vec { y } & \vec { y } \times \vec { z } & \vec { z } \times \vec { x } \end{array} \right] = \frac { 1 } { 16 } \left[ \begin{array} { l l l } \vec { u } & \vec { v } & \vec { w } \end{array} \right] ^ { 2 } \sec ^ { 2 } \frac { \alpha } { 2 } \sec ^ { 2 } \frac { \beta } { 2 } \sec ^ { 2 } \frac { \gamma } { 2 }$$