Well!why do we consider only direction cosines and not direction sines or tans. What is its actual significance?and how to use them?
Good question. According to the definition, direction cosines give the cosines of the angles which the vector makes with the positive $x,y,z$ axes. The angles range between $0$ and $180$ degrees and they give a unique direction of the vector.
If you see, $\cos x$ is continuous and takes distinct values in $[0,\pi]$ and that's what distinguishes it from $\sin$ and $\tan$.
Bcoz, when we consider either sines or tans unlike coses, we involve opposite side which depends on the plane we consider in measuring alpha, beta and gamma.
For eg: We can measure alpha in either XY or XZ plane(a plane containing X as one of its axis).
So, sin(alpha) in XY =b/a;
sin(alpha) in XZ =c/a
So involving opposite side does not give a unique line.