`vecu, vecv and vecw` are three nono-coplanar unit vectors and `alpha, beta and gamma` are the angles between `vecu and vecu, vecv and vecw and vecw and vecu`, respectively and `vecx , vecy and vecz` are unit vectors along the bisectors of the angles `alpha, beta and gamma.` respectively, prove that `[vecx xx vecy vecy xx vecz vecz xx vecx) = 1/16 [ vecu vecv vecw]^(2) sec^(2) alpha/2 sec^(2) beta/2 sec^(2) gamma/2`.

Explanation

To prove the given equation, we will follow these steps:

### Step 1: Define the Unit Vectors along the Bisectors
Let \( \vec{x}, \vec{y}, \vec{z} \) be the unit vectors along the bisectors of angles \( \alpha, \beta, \gamma \) respectively.

The unit vector along the bisector of two vectors \( \vec{u} \) and \( \vec{v} \) is given by:
\[
\vec{x} = \frac{\vec{u} + \vec{v}}{|\vec{u} + \vec{v}|}
...