Firstly,
Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle \[\alpha \].
Then, the direction cosines of the vector are \[\cos \alpha \],\[\cos \alpha \]and \[\cos \alpha \].
Now, we know that
Therefore, the direction cosines of the vector which are equally inclined to the axes are \[\frac{1}{\sqrt{3}}\],\[\frac{1}{\sqrt{3}}\],\[\frac{1}{\sqrt{3}}\].
Hence proved.