Why are these called direction angles?
Why do we consider only direction cosines and not direction sines or tans.
What is its actual significance?
And How to use them?
Why are they called direction angles?
Because direction cosines are, unlike sines and tans, even functions of the angle which makes the sign of the angle irrelevant and that's a good thing.
More importantly, the direction cosines of a unit vector $\vec v$ end up being the coordinates $v_x,v_y,v_z$, respectively, so the direction cosines obey $$\cos^2 a+\cos^2 b+\cos^2 c = 1$$ which is nice. At the end, the special feature of the cosine is that it appears in the inner product of two vectors: $$ |\vec u\cdot \vec v| = |u|\cdot |v| \cdot \cos\varphi$$ One may talk about "direction sines" or "tans" as well but they're calculable by more complicated formulae and/or obey other, more complicated identities, so they're less useful for direction calculations of interesting quantities. Sines and tans are not really banned; they are just not equally useful or natural.