A line makes 45∘ 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?

[b] We know that sum of square of direction cosines = 1$i.e.\cos 2\alpha +\cos 2\beta +\cos 2\gamma =1$⇒cos245∘+cos2β+cos2β=1$\left(Asgiven\alpha =45\circ \text{and}\beta =\gamma \right)$⇒12+2cos2β=1$\Rightarrow \cos 2\beta =14$⇒cosβ=±12, Negative value is discarded, since the line makes angle with positive axes.$Hence,\cos \beta =12$⇒cosβ=cos60∘$\beta =60\circ$∴ Required sum =α+β+γ=45∘+60∘+60∘$=165\circ$