A line makes $${45^ \circ }$$ with positive $$x$$-axis and makes equal angles with positive $$y,\, z$$  axes, respectively. What is the sum of the three angles which the line makes with positive $$x,\,y$$  and $$z$$ axes ?

Solution :
We know that sum of square of direction cosines $$ = 1$$
$$\eqalign{ & {\text{i}}{\text{.e}}{\text{., }}{\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1 \cr & \Rightarrow {\cos ^2}{45^ \circ } + {\cos ^2}\beta + {\cos ^2}\beta = 1 \cr & \left( {{\text{As given }}\alpha = {{45}^ \circ }{\text{ and }}\beta = \gamma } \right) \cr & \Rightarrow \frac{1}{2} + 2{\cos ^2}\beta = 1 \cr & \Rightarrow {\cos ^2}\beta = \frac{1}{4} \cr & \Rightarrow \cos \,\beta = \pm \frac{1}{2}, \cr} $$
Negative value is discarded,
Since the line makes angle with positive axes.
Hence, $$\cos \,\beta = \frac{1}{2} \Rightarrow \cos \,\beta = \cos \,{60^ \circ } \Rightarrow \beta = {60^ \circ }$$
$$\therefore $$  Required sum $$ = \alpha + \beta + \gamma = {45^ \circ } + {60^ \circ } + {60^ \circ } = {165^ \circ }$$